]> UoA-math-corpus

Speech and Accessibility Group Logo UoAMathCorpus: a collection of mathematical expressions in MathML

UoAMathCorpus is a collection of representative mathematical expressions in MathML developed by the Speech and Accessibility Lab., National and Kapodistrian University of Athens, Greece. It includes a section with a mix of math and text, both for the English and Greek languages. UoAMathCorpus has been designed as a research tool for eAccessiblity.

You can view UoAMathCorpus using one of the following browsers: a) Mozilla Firefox, b) Internet Explorer 11 with the MathPlayer plugin, c) Google Chrome with the extension MathJax for Chrome

Terms of Use: UoAMathCorpus is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike Licence 3.0 creative commons icon

You can download UoAMathCorpus from: Download .docx file.

The proper reference to the Corpus UoAMathCorpus that is required by the license consists of the combination of the following two references:

1.Fractions

1.1 Simple

1 1 2 + 1 3 1 4 + 1 5 =ln2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgk HiTmaalaaabaGaaGymaaqaaiaaikdaaaGaey4kaSYaaSaaaeaacaaI XaaabaGaaG4maaaacqGHsisldaWcaaqaaiaaigdaaeaacaaI0aaaai abgUcaRmaalaaabaGaaGymaaqaaiaaiwdaaaGaeyOeI0IaeS47IWKa eyypa0JaciiBaiaac6gacaaIYaaaaa@46F3@

1 1 3 + 1 5 1 7 + 1 9 = π 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgk HiTmaalaaabaGaaGymaaqaaiaaiodaaaGaey4kaSYaaSaaaeaacaaI XaaabaGaaGynaaaacqGHsisldaWcaaqaaiaaigdaaeaacaaI3aaaai abgUcaRmaalaaabaGaaGymaaqaaiaaiMdaaaGaeyOeI0IaeS47IWKa eyypa0ZaaSaaaeaacqaHapaCaeaacaaI0aaaaaaa@46E8@

1 2 1 5 + 1 8 1 11 + 1 14 = π 3 9 + 1 3 ln2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaGOmaaaacqGHsisldaWcaaqaaiaaigdaaeaacaaI1aaa aiabgUcaRmaalaaabaGaaGymaaqaaiaaiIdaaaGaeyOeI0YaaSaaae aacaaIXaaabaGaaGymaiaaigdaaaGaey4kaSYaaSaaaeaacaaIXaaa baGaaGymaiaaisdaaaGaeyOeI0IaeS47IWKaeyypa0ZaaSaaaeaacq aHapaCdaGcaaqaaiaaiodaaSqabaaakeaacaaI5aaaaiabgUcaRmaa laaabaGaaGymaaqaaiaaiodaaaGaciiBaiaac6gacaaIYaaaaa@4F15@

1 1 2 + 1 2 2 + 1 3 2 + 1 4 2 += π 2 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaGymamaaCaaaleqabaGaaGOmaaaaaaGccqGHRaWkdaWc aaqaaiaaigdaaeaacaaIYaWaaWbaaSqabeaacaaIYaaaaaaakiabgU caRmaalaaabaGaaGymaaqaaiaaiodadaahaaWcbeqaaiaaikdaaaaa aOGaey4kaSYaaSaaaeaacaaIXaaabaGaaGinamaaCaaaleqabaGaaG OmaaaaaaGccqGHRaWkcqWIVlctcqGH9aqpdaWcaaqaaiabec8aWnaa CaaaleqabaGaaGOmaaaaaOqaaiaaiAdaaaaaaa@49DD@

1 1 4 + 1 2 4 + 1 3 4 + 1 4 4 += π 4 90 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaGymamaaCaaaleqabaGaaGinaaaaaaGccqGHRaWkdaWc aaqaaiaaigdaaeaacaaIYaWaaWbaaSqabeaacaaI0aaaaaaakiabgU caRmaalaaabaGaaGymaaqaaiaaiodadaahaaWcbeqaaiaaisdaaaaa aOGaey4kaSYaaSaaaeaacaaIXaaabaGaaGinamaaCaaaleqabaGaaG inaaaaaaGccqGHRaWkcqWIVlctcqGH9aqpdaWcaaqaaiabec8aWnaa CaaaleqabaGaaGinaaaaaOqaaiaaiMdacaaIWaaaaaaa@4AA4@

1 1 6 + 1 2 6 + 1 3 6 + 1 4 6 += π 6 945 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaGymamaaCaaaleqabaGaaGOnaaaaaaGccqGHRaWkdaWc aaqaaiaaigdaaeaacaaIYaWaaWbaaSqabeaacaaI2aaaaaaakiabgU caRmaalaaabaGaaGymaaqaaiaaiodadaahaaWcbeqaaiaaiAdaaaaa aOGaey4kaSYaaSaaaeaacaaIXaaabaGaaGinamaaCaaaleqabaGaaG OnaaaaaaGccqGHRaWkcqWIVlctcqGH9aqpdaWcaaqaaiabec8aWnaa CaaaleqabaGaaGOnaaaaaOqaaiaaiMdacaaI0aGaaGynaaaaaaa@4B71@

1 1 6 + 1 2 6 + 1 3 6 + 1 4 6 += 31 π 6 30,240 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaGymamaaCaaaleqabaGaaGOnaaaaaaGccqGHRaWkdaWc aaqaaiaaigdaaeaacaaIYaWaaWbaaSqabeaacaaI2aaaaaaakiabgU caRmaalaaabaGaaGymaaqaaiaaiodadaahaaWcbeqaaiaaiAdaaaaa aOGaey4kaSYaaSaaaeaacaaIXaaabaGaaGinamaaCaaaleqabaGaaG OnaaaaaaGccqGHRaWkcqWIVlctcqGH9aqpdaWcaaqaaiaaiodacaaI XaGaeqiWda3aaWbaaSqabeaacaaI2aaaaaGcbaGaaG4maiaaicdaca GGSaGaaGOmaiaaisdacaaIWaaaaaaa@4F04@

1 1 3 + 1 3 3 1 5 3 1 7 3 += 3 π 3 2 128 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaGymamaaCaaaleqabaGaaG4maaaaaaGccqGHRaWkdaWc aaqaaiaaigdaaeaacaaIZaWaaWbaaSqabeaacaaIZaaaaaaakiabgk HiTmaalaaabaGaaGymaaqaaiaaiwdadaahaaWcbeqaaiaaiodaaaaa aOGaeyOeI0YaaSaaaeaacaaIXaaabaGaaG4namaaCaaaleqabaGaaG 4maaaaaaGccqGHRaWkcqWIVlctcqGH9aqpdaWcaaqaaiaaiodacqaH apaCdaahaaWcbeqaaiaaiodaaaGcdaGcaaqaaiaaikdaaSqabaaake aacaaIXaGaaGOmaiaaiIdaaaaaaa@4D15@

1 1×3 + 1 3×5 + 1 5×7 + 1 7×9 += 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaGymaiabgEna0kaaiodaaaGaey4kaSYaaSaaaeaacaaI XaaabaGaaG4maiabgEna0kaaiwdaaaGaey4kaSYaaSaaaeaacaaIXa aabaGaaGynaiabgEna0kaaiEdaaaGaey4kaSYaaSaaaeaacaaIXaaa baGaaG4naiabgEna0kaaiMdaaaGaey4kaSIaeS47IWKaeyypa0ZaaS aaaeaacaaIXaaabaGaaGOmaaaaaaa@4F7A@

1 1 2 × 3 2 + 1 3 2 × 5 2 + 1 5 2 × 7 2 + 1 7 2 × 9 2 += π 2 8 16 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaGymamaaCaaaleqabaGaaGOmaaaakiabgEna0kaaioda daahaaWcbeqaaiaaikdaaaaaaOGaey4kaSYaaSaaaeaacaaIXaaaba GaaG4mamaaCaaaleqabaGaaGOmaaaakiabgEna0kaaiwdadaahaaWc beqaaiaaikdaaaaaaOGaey4kaSYaaSaaaeaacaaIXaaabaGaaGynam aaCaaaleqabaGaaGOmaaaakiabgEna0kaaiEdadaahaaWcbeqaaiaa ikdaaaaaaOGaey4kaSYaaSaaaeaacaaIXaaabaGaaG4namaaCaaale qabaGaaGOmaaaakiabgEna0kaaiMdadaahaaWcbeqaaiaaikdaaaaa aOGaey4kaSIaeS47IWKaeyypa0ZaaSaaaeaacqaHapaCdaahaaWcbe qaaiaaikdaaaGccqGHsislcaaI4aaabaGaaGymaiaaiAdaaaaaaa@5B75@

β+ γ δ ε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdiMaey 4kaSYaaSaaaeaacqaHZoWzaeaacqaH0oazaaGaeyOeI0IaeqyTduga aa@3E6A@

β+γ δ+ε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq aHYoGycqGHRaWkcqaHZoWzaeaacqaH0oazcqGHRaWkcqaH1oqzaaaa aa@3E5F@

βγ δ +ε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq aHYoGycqGHsislcqaHZoWzaeaacqaH0oazaaGaey4kaSIaeqyTduga aa@3E6A@

α+β+γ+δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey 4kaSIaeqOSdiMaey4kaSIaeq4SdCMaey4kaSIaeqiTdqgaaa@3F29@

α+ β γ +δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey 4kaSYaaSaaaeaacqaHYoGyaeaacqaHZoWzaaGaey4kaSIaeqiTdqga aa@3E57@

α β+γ+δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq aHXoqyaeaacqaHYoGycqGHRaWkcqaHZoWzcqGHRaWkcqaH0oazaaaa aa@3E57@

α+β γ+δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq aHXoqycqGHRaWkcqaHYoGyaeaacqaHZoWzcqGHRaWkcqaH0oazaaaa aa@3E57@

( α+β )( γ+δ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aHXoqycqGHRaWkcqaHYoGyaiaawIcacaGLPaaacqGHflY1daqadaqa aiabeo7aNjabgUcaRiabes7aKbGaayjkaiaawMcaaaaa@43A3@

χ+ ψ 2 κ+1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq aHhpWycqGHRaWkcqaHipqEdaahaaWcbeqaaiaaikdaaaaakeaacqaH 6oWAcqGHRaWkcaaIXaaaaaaa@3EB0@

χ+ ψ 2 κ +1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4XdmMaey 4kaSYaaSaaaeaacqaHipqEdaahaaWcbeqaaiaaikdaaaaakeaacqaH 6oWAaaGaey4kaSIaaGymaaaa@3EB0@

β×( γ+δ ) ε 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq aHYoGycqGHxdaTdaqadaqaaiabeo7aNjabgUcaRiabes7aKbGaayjk aiaawMcaaaqaaiabew7aLnaaCaaaleqabaGaaGOmaaaaaaaaaa@4206@

1.2 Complex

1 1 2π + 1 3 2π + 1 5 2π + 1 7 2π += ( 2 2π 1 ) π 2π Β π 2( 2π )! MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaGymamaaCaaaleqabaGaaGOmaiabec8aWbaaaaGccqGH RaWkdaWcaaqaaiaaigdaaeaacaaIZaWaaWbaaSqabeaacaaIYaGaeq iWdahaaaaakiabgUcaRmaalaaabaGaaGymaaqaaiaaiwdadaahaaWc beqaaiaaikdacqaHapaCaaaaaOGaey4kaSYaaSaaaeaacaaIXaaaba GaaG4namaaCaaaleqabaGaaGOmaiabec8aWbaaaaGccqGHRaWkcqWI VlctcqGH9aqpdaWcaaqaamaabmaabaGaaGOmamaaCaaaleqabaGaaG Omaiabec8aWbaakiabgkHiTiaaigdaaiaawIcacaGLPaaacqaHapaC daahaaWcbeqaaiaaikdacqaHapaCaaGccqqHsoGqdaWgaaWcbaGaeq iWdahabeaaaOqaaiaaikdadaqadaqaaiaaikdacqaHapaCaiaawIca caGLPaaacaGGHaaaaaaa@6129@

1 1 2π 1 2 2π + 1 3 2π 1 4 2π += ( 2 2π1 1 ) π 2π Β π ( 2π )! MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaGymamaaCaaaleqabaGaaGOmaiabec8aWbaaaaGccqGH sisldaWcaaqaaiaaigdaaeaacaaIYaWaaWbaaSqabeaacaaIYaGaeq iWdahaaaaakiabgUcaRmaalaaabaGaaGymaaqaaiaaiodadaahaaWc beqaaiaaikdacqaHapaCaaaaaOGaeyOeI0YaaSaaaeaacaaIXaaaba GaaGinamaaCaaaleqabaGaaGOmaiabec8aWbaaaaGccqGHRaWkcqWI VlctcqGH9aqpdaWcaaqaamaabmaabaGaaGOmamaaCaaaleqabaGaaG Omaiabec8aWjabgkHiTiaaigdaaaGccqGHsislcaaIXaaacaGLOaGa ayzkaaGaeqiWda3aaWbaaSqabeaacaaIYaGaeqiWdahaaOGaeuOKdi 0aaSbaaSqaaiabec8aWbqabaaakeaadaqadaqaaiaaikdacqaHapaC aiaawIcacaGLPaaacaGGHaaaaaaa@6225@

1 1 2π+1 1 3 2π+1 + 1 5 2π+1 1 7 2π+1 += π 2π+1 Ε π 2 2π+2 ( 2π )! MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaGymamaaCaaaleqabaGaaGOmaiabec8aWjabgUcaRiaa igdaaaaaaOGaeyOeI0YaaSaaaeaacaaIXaaabaGaaG4mamaaCaaale qabaGaaGOmaiabec8aWjabgUcaRiaaigdaaaaaaOGaey4kaSYaaSaa aeaacaaIXaaabaGaaGynamaaCaaaleqabaGaaGOmaiabec8aWjabgU caRiaaigdaaaaaaOGaeyOeI0YaaSaaaeaacaaIXaaabaGaaG4namaa CaaaleqabaGaaGOmaiabec8aWjabgUcaRiaaigdaaaaaaOGaey4kaS IaeS47IWKaeyypa0ZaaSaaaeaacqaHapaCdaahaaWcbeqaaiaaikda cqaHapaCcqGHRaWkcaaIXaaaaOGaeuyLdu0aaSbaaSqaaiabec8aWb qabaaakeaacaaIYaWaaWbaaSqabeaacaaIYaGaeqiWdaNaey4kaSIa aGOmaaaakmaabmaabaGaaGOmaiabec8aWbGaayjkaiaawMcaaiaacg caaaaaaa@6707@

( α+χ ) ν = α ν +ν α ν1 χ+ ν(ν1) 2! α ν2 χ 2 + ν(ν1)(ν2) 3! α ν3 χ 3 + MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaqada qaaiabeg7aHjabgUcaRiabeE8aJbGaayjkaiaawMcaamaaCaaaleqa baGaeqyVd4gaaOGaeyypa0JaeqySde2aaWbaaSqabeaacqaH9oGBaa GccqGHRaWkcqaH9oGBcqaHXoqydaahaaWcbeqaaiabe27aUjabgkHi TiaaigdaaaGccqaHhpWycqGHRaWkdaWcaaqaaiabe27aUjaacIcacq aH9oGBcqGHsislcaaIXaGaaiykaaqaaiaaikdacaGGHaaaaiabeg7a HnaaCaaaleqabaGaeqyVd4MaeyOeI0IaaGOmaaaakiabeE8aJnaaCa aaleqabaGaaGOmaaaaaOqaaiabgUcaRmaalaaabaGaeqyVd4Maaiik aiabe27aUjabgkHiTiaaigdacaGGPaGaaiikaiabe27aUjabgkHiTi aaikdacaGGPaaabaGaaG4maiaacgcaaaGaeqySde2aaWbaaSqabeaa cqaH9oGBcqGHsislcaaIZaaaaOGaeq4Xdm2aaWbaaSqabeaacaaIZa aaaOGaey4kaSIaeS47IWeaaaa@74A0@

α χ = e χlnα =1+χlnα+ ( χlnα ) 2 2! + ( χlnα ) 3 3! + MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaW baaSqabeaacqaHhpWyaaGccqGH9aqpcaWGLbWaaWbaaSqabeaacqaH hpWyciGGSbGaaiOBaiabeg7aHbaakiabg2da9iaaigdacqGHRaWkcq aHhpWyciGGSbGaaiOBaiabeg7aHjabgUcaRmaalaaabaWaaeWaaeaa cqaHhpWyciGGSbGaaiOBaiabeg7aHbGaayjkaiaawMcaamaaCaaale qabaGaaGOmaaaaaOqaaiaaikdacaGGHaaaaiabgUcaRmaalaaabaWa aeWaaeaacqaHhpWyciGGSbGaaiOBaiabeg7aHbGaayjkaiaawMcaam aaCaaaleqabaGaaG4maaaaaOqaaiaaiodacaGGHaaaaiabgUcaRiab l+Uimbaa@5FA6@

lnχ=2{ ( χ1 χ+1 )+ 1 3 ( χ1 χ+1 ) 3 + 1 5 ( χ1 χ+1 ) 5 + } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac6 gacqaHhpWycqGH9aqpcaaIYaWaaiWaaeaadaqadaqaamaalaaabaGa eq4XdmMaeyOeI0IaaGymaaqaaiabeE8aJjabgUcaRiaaigdaaaaaca GLOaGaayzkaaGaey4kaSYaaSaaaeaacaaIXaaabaGaaG4maaaadaqa daqaamaalaaabaGaeq4XdmMaeyOeI0IaaGymaaqaaiabeE8aJjabgU caRiaaigdaaaaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIZaaaaOGa ey4kaSYaaSaaaeaacaaIXaaabaGaaGynaaaadaqadaqaamaalaaaba Gaeq4XdmMaeyOeI0IaaGymaaqaaiabeE8aJjabgUcaRiaaigdaaaaa caGLOaGaayzkaaWaaWbaaSqabeaacaaI1aaaaOGaey4kaSIaeS47IW eacaGL7bGaayzFaaaaaa@5FF9@

2sinμπ π ( sinχ 1 μ 2 2sin2χ 2 2 μ 2 + 3sin3χ 3 2 μ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIYaGaci4CaiaacMgacaGGUbGaeqiVd0MaeqiWdahabaGaeqiWdaha amaabmaabaWaaSaaaeaaciGGZbGaaiyAaiaac6gacqaHhpWyaeaaca aIXaGaeyOeI0IaeqiVd02aaWbaaSqabeaacaaIYaaaaaaakiabgkHi TmaalaaabaGaaGOmaiGacohacaGGPbGaaiOBaiaaikdacqaHhpWyae aacaaIYaWaaWbaaSqabeaacaaIYaaaaOGaeyOeI0IaeqiVd02aaWba aSqabeaacaaIYaaaaaaakiabgUcaRmaalaaabaGaaG4maiGacohaca GGPbGaaiOBaiaaiodacqaHhpWyaeaacaaIZaWaaWbaaSqabeaacaaI YaaaaOGaeyOeI0IaeqiVd02aaWbaaSqabeaacaaIYaaaaaaakiabgk HiTiabl+UimbGaayjkaiaawMcaaaaa@64A9@

1+ χ 1+ χ 1+ χ 1+ χ 1+ χ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgU caRmaalaaabaGaeq4XdmgabaGaaGymaiabgUcaRmaalaaabaGaeq4X dmgabaGaaGymaiabgUcaRmaalaaabaGaeq4XdmgabaGaaGymaiabgU caRmaalaaabaGaeq4XdmgabaGaaGymaiabgUcaRmaalaaabaGaeq4X dmgabaGaeSOjGSeaaaaaaaaaaaaaaaa@480D@

φ΄(χ)+φ(χ)×ψ+ζ χ×ψ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq aHgpGAcaWGeoGaaiikaiabeE8aJjaacMcacqGHRaWkcqaHgpGAcaGG OaGaeq4XdmMaaiykaiabgEna0kabeI8a5jabgUcaRiabeA7a6bqaai abeE8aJjabgEna0kabeI8a5baaaaa@4DAF@

χ= β± β 2 4αγ 2α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4XdmMaey ypa0ZaaSaaaeaacqGHsislcqaHYoGycqGHXcqSdaGcaaqaaiabek7a InaaCaaaleqabaGaaGOmaaaakiabgkHiTiaaisdacqGHflY1cqaHXo qycqGHflY1cqaHZoWzaSqabaaakeaacaaIYaGaeyyXICTaeqySdega aaaa@4E23@

2. Roots

2.1 Simple

β γ +δε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaada Wcaaqaaiabek7aIbqaaiabeo7aNbaaaSqabaGccqGHRaWkcqaH0oaz cqGHsislcqaH1oqzaaa@3E8F@

β γ+δ ε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaada Wcaaqaaiabek7aIbqaaiabeo7aNjabgUcaRiabes7aKbaaaSqabaGc cqGHsislcqaH1oqzaaa@3E8F@

β γ ε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaada Wcaaqaaiabek7aIbqaaiabeo7aNbaacqGHsislcqaH1oqzaSqabaaa aa@3BFE@

β+γ + δ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacq aHYoGycqGHRaWkcqaHZoWzaSqabaGccqGHRaWkcqaH0oazdaahaaWc beqaaiaaikdaaaaaaa@3DB6@

β +γ δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaada Gcaaqaaiabek7aIbWcbeaakiabgUcaRiabeo7aNbqaaiabes7aKbaa aaa@3BFB@

2.2 Complex

π 2 π 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaada Gcaaqaaiabec8aWbWcbeaaaOqaaiaaikdaaaGaeyiyIK7aaOaaaeaa daWcaaqaaiabec8aWbqaaiaaikdaaaaaleqaaaaa@3D10@

1+ 2+ 2+ 2+ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aIXaGaey4kaSYaaOaaaeaacaaIYaGaey4kaSYaaOaaaeaacaaIYaGa ey4kaSYaaOaaaeaacaaIYaGaey4kaSIaeSOjGSealeqaaaqabaaabe aaaeqaaaaa@3DDB@

d( χ,ψ )= ( χ 1 ψ 1 ) 2 + ( χ 2 ψ 2 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaabm aabaGaeq4XdmMaaiilaiabeI8a5bGaayjkaiaawMcaaiabg2da9maa kaaabaWaaeWaaeaacqaHhpWydaWgaaWcbaGaaGymaaqabaGccqGHsi slcqaHipqEdaWgaaWcbaGaaGymaaqabaaakiaawIcacaGLPaaadaah aaWcbeqaaiaaikdaaaGccqGHRaWkdaqadaqaaiabeE8aJnaaBaaale aacaaIYaaabeaakiabgkHiTiabeI8a5naaBaaaleaacaaIYaaabeaa aOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaeqaaaaa@502E@

Σ= α β 2π×φ(χ)× 1+ [ φ'(χ) ] 2 dχ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4OdmLaey ypa0Zaa8qmaeaacaaIYaGaeqiWdaNaey41aqRaeqOXdOMaaiikaiab eE8aJjaacMcacqGHxdaTdaGcaaqaaiaaigdacqGHRaWkdaWadaqaai abeA8aQjaacEcacaGGOaGaeq4XdmMaaiykaaGaay5waiaaw2faamaa CaaaleqabaGaaGOmaaaaaeqaaOGaamizaiabeE8aJbWcbaGaeqySde gabaGaeqOSdiganiabgUIiYdaaaa@5608@

σ= [ ( χ χ ¯ ) 2 ν ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaey ypa0ZaaOaaaeaadaWadaqaamaalaaabaWaaabqaeaadaqadaqaaiab eE8aJjabgkHiTmaanaaabaGaeq4XdmgaaaGaayjkaiaawMcaamaaCa aaleqabaGaaGOmaaaaaeqabeqdcqGHris5aaGcbaGaeqyVd4gaaaGa ay5waiaaw2faaaWcbeaaaaa@4586@

3. Derivatives

Simple

Ω= Ω ¯ Ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOaIyRaeu yQdCLaeyypa0ZaaSaaaeaacuqHPoWvgaqeaaqaaiabfM6axbaaaaa@3D35@

υ τ ( χ,τ )=( Λ( χ )υ( χ,τ ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyXdu3aaS baaSqaaiabes8a0bqabaGcdaqadaqaaiabeE8aJjaacYcacqaHepaD aiaawIcacaGLPaaacqGH9aqpcqGHhis0daqadaqaaiabfU5amnaabm aabaGaeq4XdmgacaGLOaGaayzkaaGaey4bIeTaeqyXdu3aaeWaaeaa cqaHhpWycaGGSaGaeqiXdqhacaGLOaGaayzkaaaacaGLOaGaayzkaa aaaa@513A@

Δ υ( χ )= Κ,σ υ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4bIe9aaS baaSqaaiabfs5aebqabaGccqaHfpqDdaqadaqaaiabeE8aJbGaayjk aiaawMcaaiabg2da9iabgEGirpaaBaaaleaacqqHAoWscaGGSaGaeq 4Wdmhabeaakiabew8a1baa@468F@

0 υ ν = υ ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOaIy7aaW baaSqabeaacaaIWaaaaOGaeqyXdu3aaSbaaSqaaiabe27aUbqabaGc cqGH9aqpcqaHfpqDdaWgaaWcbaGaeqyVd4gabeaaaaa@40B4@

2 υ ν = 1 κ ( 1 υ ν 1 υ ν1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOaIy7aaW baaSqabeaacaaIYaaaaOGaeqyXdu3aaSbaaSqaaiabe27aUbqabaGc cqGH9aqpdaWcaaqaaiaaigdaaeaacqaH6oWAaaWaaeWaaeaacqGHci ITdaahaaWcbeqaaiaaigdaaaGccqaHfpqDdaWgaaWcbaGaeqyVd4ga beaakiabgkHiTiabgkGi2oaaCaaaleqabaGaaGymaaaakiabew8a1n aaBaaaleaacqaH9oGBcqGHsislcaaIXaaabeaaaOGaayjkaiaawMca aaaa@4FC0@

α Δ η Δ ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aau WaaeaacqGHhis0daWgaaWcbaGaeuiLdqeabeaakiabeE7aOnaaDaaa leaacqqHuoaraeaacqaH9oGBaaaakiaawMa7caGLkWoaaaa@42E0@

d 2 υ d λ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGKbWaaWbaaSqabeaacaaIYaaaaOGaeqyXduhabaGaamizaiabeU7a SnaaCaaaleqabaGaaGOmaaaaaaaaaa@3D30@

( Φ( τ,υ( τ ) ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4bIe9aae WaaeaacqqHMoGrdaqadaqaaiabes8a0jaacYcacqaHfpqDdaqadaqa aiabes8a0bGaayjkaiaawMcaaaGaayjkaiaawMcaaaGaayjkaiaawM caaaaa@4393@

υ ¨ ( τ )=Φ( τ,υ( τ ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyXduNbam aadaqadaqaaiabes8a0bGaayjkaiaawMcaaiabg2da9iabgEGirlab fA6agnaabmaabaGaeqiXdqNaaiilaiabew8a1naabmaabaGaeqiXdq hacaGLOaGaayzkaaaacaGLOaGaayzkaaaaaa@482F@

| Η( τ,χ ) |φ( τ ) | χ | α +γ( τ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacq GHhis0cqqHxoasdaqadaqaaiabes8a0jaacYcacqaHhpWyaiaawIca caGLPaaaaiaawEa7caGLiWoacqGHKjYOcqaHgpGAdaqadaqaaiabes 8a0bGaayjkaiaawMcaamaaemaabaGaeq4XdmgacaGLhWUaayjcSdWa aWbaaSqabeaacqaHXoqyaaGccqGHRaWkcqaHZoWzdaqadaqaaiabes 8a0bGaayjkaiaawMcaaaaa@5507@

φ ( υ ν )0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqOXdOMbau aadaqadaqaaiabew8a1naaBaaaleaacqaH9oGBaeqaaaGccaGLOaGa ayzkaaGaeyOKH4QaaGimaaaa@3FA5@

υ ˜ ν | φ ( υ ν ), υ ˜ ν | MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaauWaaeaacu aHfpqDgaacamaaBaaaleaacqaH9oGBaeqaaaGccaGLjWUaayPcSdGa eyyzIm7aaqWaaeaadaaadaqaaiqbeA8aQzaafaWaaeWaaeaacqaHfp qDdaWgaaWcbaGaeqyVd4gabeaaaOGaayjkaiaawMcaaiaacYcacuaH fpqDgaacamaaBaaaleaacqaH9oGBaeqaaaGccaGLPmIaayPkJaaaca GLhWUaayjcSdaaaa@4F15@

ξ υ 1 τ ξ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHciITdaahaaWcbeqaaiabe67a4baakiabew8a1naaBaaaleaacaaI XaaabeaaaOqaaiabgkGi2kabes8a0naaCaaaleqabaGaeqOVdGhaaa aaaaa@413A@

ιΔυ υ τ φ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyUdKMaey iLdqKaeqyXduNaeyOeI0IaeqyXdu3aaSbaaSqaaiabes8a0bqabaGc cqaHgpGAaaa@4140@

Δ ω υ+κ×υ=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiLdq0aaS baaSqaaiabeM8a3bqabaGccqaHfpqDcqGHRaWkcqaH6oWAcqGHxdaT cqaHfpqDcqGH9aqpcaaIWaaaaa@435A@

χ=ρ× χ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4XdmMaey ypa0JaeqyWdiNaey41aqRafq4XdmMbauaaaaa@3E4E@

υ ω | ω=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqGaaeaada WcaaqaaiabgkGi2kabew8a1bqaaiabgkGi2kabeM8a3baaaiaawIa7 amaaBaaaleaacqaHjpWDcqGH9aqpcaaIWaaabeaaaaa@41B6@

2 υ v 2 λ 2 υ=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHciITdaahaaWcbeqaaiaaikdaaaGccqaHfpqDaeaacqGHciITcaWG 2bWaaWbaaSqabeaacaaIYaaaaaaakiabgkHiTiabeU7aSnaaCaaale qabaGaaGOmaaaakiabew8a1jabg2da9iaaicdaaaa@4496@

χ ξ γ 2 ( τ )lnρ= γ 2 ( τ ) χ ξ ρ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHciITaeaacqGHciITcqaHhpWydaWgaaWcbaGaeqOVdGhabeaaaaGc cqaHZoWzdaWgaaWcbaGaaGOmaaqabaGcdaqadaqaaiabes8a0bGaay jkaiaawMcaaiGacYgacaGGUbGaeqyWdiNaeyypa0Jaeq4SdC2aaSba aSqaaiaaikdaaeqaaOWaaeWaaeaacqaHepaDaiaawIcacaGLPaaada WcaaqaaiabeE8aJnaaBaaaleaacqaH+oaEaeqaaaGcbaGaeqyWdi3a aWbaaSqabeaacaaIYaaaaaaaaaa@5364@

1 ρ sin ωπ ω 0 ω MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaeqyWdihaaiGacohacaGGPbGaaiOBamaalaaabaGaeqyY dCNaeqiWdahabaGaeqyYdC3aaSbaaSqaaiaaicdaaeqaaaaakmaala aabaGaeyOaIylabaGaeyOaIyRaeqyYdChaaaaa@465A@

υ( 0 )= υ ( 1 )= υ ( 0 )= υ ( 1 )=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyXdu3aae WaaeaacaaIWaaacaGLOaGaayzkaaGaeyypa0JafqyXduNbauaadaqa daqaaiaaigdaaiaawIcacaGLPaaacqGH9aqpcuaHfpqDgaGbamaabm aabaGaaGimaaGaayjkaiaawMcaaiabg2da9iqbew8a1zaasaWaaeWa aeaacaaIXaaacaGLOaGaayzkaaGaeyypa0JaaGimaaaa@4B25@

υ ( 4 ) ( τ )=φ( τ,υ( τ ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyXdu3aaW baaSqabeaadaqadaqaaiaaisdaaiaawIcacaGLPaaaaaGcdaqadaqa aiabes8a0bGaayjkaiaawMcaaiabg2da9iabeA8aQnaabmaabaGaeq iXdqNaaiilaiabew8a1naabmaabaGaeqiXdqhacaGLOaGaayzkaaaa caGLOaGaayzkaaaaaa@4960@

2 u 2 Ξ( τ,σ )=Γ( τ,σ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHciITdaahaaWcbeqaaiaaikdaaaaakeaacqGHciITcaWG1bWaaWba aSqabeaacaaIYaaaaaaakiabf65aynaabmaabaGaeqiXdqNaaiilai abeo8aZbGaayjkaiaawMcaaiabg2da9iabgkHiTiabfo5ahnaabmaa baGaeqiXdqNaaiilaiabeo8aZbGaayjkaiaawMcaaaaa@4C14@

| υ |=max{ υ , υ , υ } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaada qbdaqaaiabew8a1bGaayzcSlaawQa7aaGaay5bSlaawIa7aiabg2da 9iGac2gacaGGHbGaaiiEamaacmaabaWaauWaaeaacqaHfpqDaiaawM a7caGLkWoacaGGSaWaauWaaeaacuaHfpqDgaqbaaGaayzcSlaawQa7 aiaacYcadaqbdaqaaiqbew8a1zaagaaacaGLjWUaayPcSdaacaGL7b GaayzFaaaaaa@5455@

( υ * ) ( τ )= ( Τ υ * ) ( τ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aHfpqDdaahaaWcbeqaaiaacQcaaaaakiaawIcacaGLPaaadaahaaWc beqaaOGamai4gkdiIUGaaGzaVRGamai4gkdiIcaadaqadaqaaiabes 8a0bGaayjkaiaawMcaaiabg2da9maabmaabaGaeuiPdqLaeqyXdu3a aWbaaSqabeaacaGGQaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaaki adacUHYaIOliaaygW7kiadacUHYaIOaaWaaeWaaeaacqaHepaDaiaa wIcacaGLPaaaaaa@56D5@

3.2 Complex

d dτ ( | υ ˙ ( τ ) | π( τ )2 υ ˙ ( τ ) )=Φ( τ,υ( τ ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGKbaabaGaamizaiabes8a0baadaqadaqaamaaemaabaGafqyXduNb aiaadaqadaqaaiabes8a0bGaayjkaiaawMcaaaGaay5bSlaawIa7am aaCaaaleqabaGaeqiWda3aaeWaaeaacqaHepaDaiaawIcacaGLPaaa cqGHsislcaaIYaaaaOGafqyXduNbaiaadaqadaqaaiabes8a0bGaay jkaiaawMcaaaGaayjkaiaawMcaaiabg2da9iabgEGirlabfA6agnaa bmaabaGaeqiXdqNaaiilaiabew8a1naabmaabaGaeqiXdqhacaGLOa GaayzkaaaacaGLOaGaayzkaaaaaa@5C89@

d dτ ( υ 2 +2 Ω Φ( υ )dχ )+2 υ τ 2 =2( ( α τ , υ τ ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWcaa qaaiaadsgaaeaacaWGKbGaeqiXdqhaamaabmaabaWaauWaaeaacqGH his0cqaHfpqDaiaawMa7caGLkWoadaahaaWcbeqaaiaaikdaaaGccq GHRaWkcaaIYaWaa8quaeaacqqHMoGrdaqadaqaaiabew8a1bGaayjk aiaawMcaaiaadsgacqaHhpWyaSqaaiabfM6axbqab0Gaey4kIipaaO GaayjkaiaawMcaaiabgUcaRiaaikdadaqbdaqaamaalaaabaGaeyOa IyRaeqyXduhabaGaeyOaIyRaeqiXdqhaaaGaayzcSlaawQa7amaaCa aaleqabaGaaGOmaaaaaOqaaiabg2da9iaaikdadaqadaqaamaabmaa baWaaSaaaeaacqGHciITcqaHXoqyaeaacqGHciITcqaHepaDaaGaai ilamaalaaabaGaeyOaIyRaeqyXduhabaGaeyOaIyRaeqiXdqhaaaGa ayjkaiaawMcaaaGaayjkaiaawMcaaaaaaa@6DCB@

τ ( υ τ )Δ υ τ + φ ( υ ) υ τ = α τ +Δαυ υ τ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWcaa qaaiabgkGi2cqaaiabgkGi2kabes8a0baadaqadaqaamaalaaabaGa eyOaIyRaeqyXduhabaGaeyOaIyRaeqiXdqhaaaGaayjkaiaawMcaai abgkHiTiabgs5aenaalaaabaGaeyOaIyRaeqyXduhabaGaeyOaIyRa eqiXdqhaaiabgUcaRiqbeA8aQzaafaWaaeWaaeaacqaHfpqDaiaawI cacaGLPaaadaWcaaqaaiabgkGi2kabew8a1bqaaiabgkGi2kabes8a 0baaaeaacqGH9aqpcqGHsisldaWcaaqaaiabgkGi2kabeg7aHbqaai abgkGi2kabes8a0baacqGHRaWkcqGHuoarcqaHXoqycqGHsislcqaH fpqDcqGHsisldaWcaaqaaiabgkGi2kabew8a1bqaaiabgkGi2kabes 8a0baaaaaa@6DD3@

d dτ ( Α 3 2 α 2 + Α dα dτ 2 )γ Δυ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGKbaabaGaamizaiabes8a0baadaqadaqaamaafmaabaGaeuyKde0a aWbaaSqabeaadaWcaaqaaiaaiodaaeaacaaIYaaaaaaakiabeg7aHb GaayzcSlaawQa7amaaCaaaleqabaGaaGOmaaaakiabgUcaRmaafmaa baGaeuyKde0aaSaaaeaacaWGKbGaeqySdegabaGaamizaiabes8a0b aaaiaawMa7caGLkWoadaahaaWcbeqaaiaaikdaaaaakiaawIcacaGL PaaacqGHKjYOcqaHZoWzdaqbdaqaaiabgs5aejabew8a1bGaayzcSl aawQa7amaaCaaaleqabaGaaGOmaaaaaaa@5A3C@

υ μ τ + ν μ τ + υ ¯ μ χ = φ ¯ μ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHciITcqaHfpqDdaWgaaWcbaGaeqiVd0gabeaaaOqaaiabgkGi2kab es8a0baacqGHRaWkdaWcaaqaaiabgkGi2kabe27aUnaaBaaaleaacq aH8oqBaeqaaaGcbaGaeyOaIyRaeqiXdqhaaiabgUcaRmaalaaabaGa eyOaIyRafqyXduNbaebadaWgaaWcbaGaeqiVd0gabeaaaOqaaiabgk Gi2kabeE8aJbaacqGH9aqpcuaHgpGAgaqeamaaBaaaleaacqaH8oqB aeqaaaaa@556F@

sup 0τΤ [ ( υ ν υ ) τ ]+ sup 0τΤ [ ( θ ν θ ) τ ] Γ( λ ν+1 1 8 + μ ν+1 1 8 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWfqa qaaiGacohacaGG1bGaaiiCaaWcbaGaaGimaiabgsMiJkabes8a0jab gsMiJkabfs6aubqabaGcdaWadaqaamaalaaabaGaeyOaIy7aaeWaae aacqaHfpqDdaWgaaWcbaGaeqyVd4gabeaakiabgkHiTiabew8a1bGa ayjkaiaawMcaaaqaaiabgkGi2kabes8a0baaaiaawUfacaGLDbaacq GHRaWkdaWfqaqaaiGacohacaGG1bGaaiiCaaWcbaGaaGimaiabgsMi Jkabes8a0jabgsMiJkabfs6aubqabaGcdaWadaqaamaalaaabaGaey OaIy7aaeWaaeaacqaH4oqCdaWgaaWcbaGaeqyVd4gabeaakiabgkHi TiabeI7aXbGaayjkaiaawMcaaaqaaiabgkGi2kabes8a0baaaiaawU facaGLDbaaaeaacqGHKjYOcqqHtoWrdaqadaqaaiabeU7aSnaaDaaa leaacqaH9oGBcqGHRaWkcaaIXaaabaGaeyOeI0YaaSaaaeaacaaIXa aabaGaaGioaaaaaaGccqGHRaWkcqaH8oqBdaqhaaWcbaGaeqyVd4Ma ey4kaSIaaGymaaqaaiabgkHiTmaalaaabaGaaGymaaqaaiaaiIdaaa aaaaGccaGLOaGaayzkaaaaaaa@7DF6@

d[ φ(χ) τ(χ) ] dx = τ(χ)× d[ φ(χ) ] dx φ(χ)× d[ τ(χ) ] dx [ τ(χ) ] 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGKbWaamWaaeaadaWcaaqaaiabeA8aQjaacIcacqaHhpWycaGGPaaa baGaeqiXdqNaaiikaiabeE8aJjaacMcaaaaacaGLBbGaayzxaaaaba GaamizaiaadIhaaaGaeyypa0ZaaSaaaeaacqaHepaDcaGGOaGaeq4X dmMaaiykaiabgEna0oaalaaabaGaamizamaadmaabaGaeqOXdOMaai ikaiabeE8aJjaacMcaaiaawUfacaGLDbaaaeaacaWGKbGaamiEaaaa cqGHsislcqaHgpGAcaGGOaGaeq4XdmMaaiykaiabgEna0oaalaaaba GaamizamaadmaabaGaeqiXdqNaaiikaiabeE8aJjaacMcaaiaawUfa caGLDbaaaeaacaWGKbGaamiEaaaaaeaadaWadaqaaiabes8a0jaacI cacqaHhpWycaGGPaaacaGLBbGaayzxaaWaaWbaaSqabeaacaaIYaaa aaaaaaa@6F41@

4. Exponentials & Indicators

4.1 Simple

δ Κ,σ >0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS baaSqaaiabfQ5aljaacYcacqaHdpWCaeqaaOGaeyOpa4JaaGimaaaa @3D7A@

β σ Κ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOSdi2aa0 baaSqaaiabeo8aZbqaaiabfQ5albaakiabgcMi5kaaicdaaaa@3D86@

χ σ = ΚΜ β σ Κ χ Κ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaS baaSqaaiabeo8aZbqabaGccqGH9aqpdaaeqbqaaiabek7aInaaDaaa leaacqaHdpWCaeaacqqHAoWsaaGccqaHhpWydaWgaaWcbaGaeuOMdS eabeaaaeaacqqHAoWscqGHiiIZcqqHCoqtaeqaniabggHiLdaaaa@4997@

{ χ Κ } Κ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacq aHhpWydaWgaaWcbaGaeuOMdSeabeaaaOGaay5Eaiaaw2haamaaBaaa leaacqqHAoWsaeqaaaaa@3D27@

υ Κ =φ( χ Κ ),ΚΜ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyXdu3aaS baaSqaaiabfQ5albqabaGccqGH9aqpcqaHgpGAdaqadaqaaiabeE8a JnaaBaaaleaacqqHAoWsaeqaaaGccaGLOaGaayzkaaGaaiilaiabgc GiIiabfQ5aljabgIGiolabfY5anbaa@4701@

1 υ ν = υ ν υ ν1 κ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOaIy7aaW baaSqabeaacaaIXaaaaOGaeqyXdu3aaWbaaSqabeaacqaH9oGBaaGc cqGH9aqpdaWcaaqaaiabew8a1naaCaaaleqabaGaeqyVd4gaaOGaey OeI0IaeqyXdu3aaWbaaSqabeaacqaH9oGBcqGHsislcaaIXaaaaaGc baGaeqOUdSgaaaaa@48CE@

Φ Κ,σ ( υ )= σ Ε κ Α σ σ ( υ κ υ σ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOPdy0aaS baaSqaaiabfQ5aljaacYcacqaHdpWCaeqaaOWaaeWaaeaacqaHfpqD aiaawIcacaGLPaaacqGH9aqpdaaeqbqaaiabfg5abnaaCaaaleqaba Gaeq4WdmNafq4WdmNbauaaaaGcdaqadaqaaiabew8a1naaBaaaleaa cqaH6oWAaeqaaOGaeyOeI0IaeqyXdu3aaSbaaSqaaiqbeo8aZzaafa aabeaaaOGaayjkaiaawMcaaaWcbaGafq4WdmNbauaacqGHiiIZcqqH voqrdaWgaaadbaGaeqOUdSgabeaaaSqab0GaeyyeIuoaaaa@57C5@

α | υ | χ 2 υ,υ Φ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaq WaaeaacqaHfpqDaiaawEa7caGLiWoadaqhaaWcbaGaeq4XdmgabaGa aGOmaaaakiabgsMiJoaaamaabaGaeqyXduNaaiilaiabew8a1bGaay zkJiaawQYiamaaBaaaleaacqqHMoGraeqaaaaa@4892@

Γ 6 = δ 3 θ+ δ 7 2 θ 2 + δ 5 2 θ+1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4KdC0aaS baaSqaaiaaiAdaaeqaaOGaeyypa0JaeqiTdq2aaWbaaSqabeaacaaI ZaaaaOGaeqiUdeNaey4kaSIaeqiTdq2aaWbaaSqabeaadaWcaaqaai aaiEdaaeaacaaIYaaaaaaakiabeI7aXnaaCaaaleqabaGaaGOmaaaa kiabgUcaRiabes7aKnaaCaaaleqabaWaaSaaaeaacaaI1aaabaGaaG OmaaaaaaGccqaH4oqCcqGHRaWkcaaIXaaaaa@4C3A@

υ ¯ Δ 0 = υ Δ 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqyXduNbae badaqhaaWcbaGaeuiLdqeabaGaaGimaaaakiabg2da9iabew8a1naa DaaaleaacqqHuoaraeaacaaIWaaaaaaa@3F47@

θ 2 υ ν = κ 1 ( 1 υ ν 1 υ ν1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUde3aaW baaSqabeaacaaIYaaaaOGaeqyXdu3aaWbaaSqabeaacqaH9oGBaaGc cqGH9aqpcqaH6oWAdaahaaWcbeqaaiabgkHiTiaaigdaaaGcdaqada qaaiabgkGi2oaaCaaaleqabaGaaGymaaaakiabew8a1naaCaaaleqa baGaeqyVd4gaaOGaeyOeI0IaeyOaIy7aaWbaaSqabeaacaaIXaaaaO GaeqyXdu3aaWbaaSqabeaacqaH9oGBcqGHsislcaaIXaaaaaGccaGL OaGaayzkaaaaaa@5127@

χ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaauWaaeaacq aHhpWyaiaawMa7caGLkWoaaaa@3AD5@

α η Δ ν+1 χ Γ π Μ Δ ν+1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aau WaaeaacqaH3oaAdaqhaaWcbaGaeuiLdqeabaGaeqyVd4Maey4kaSIa aGymaaaaaOGaayzcSlaawQa7aiabeE8aJjabgsMiJkabfo5ahnaaBa aaleaacqaHapaCaeqaaOGaeuiNd00aa0baaSqaaiabfs5aebqaaiab e27aUjabgUcaRiaaigdaaaaaaa@4E81@

Φ 2 ( τ,χ )=| sinωτ | | χ | 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOPdy0aaS baaSqaaiaaikdaaeqaaOWaaeWaaeaacqaHepaDcaGGSaGaeq4Xdmga caGLOaGaayzkaaGaeyypa0ZaaqWaaeaaciGGZbGaaiyAaiaac6gacq aHjpWDcqaHepaDaiaawEa7caGLiWoadaabdaqaaiabeE8aJbGaay5b SlaawIa7amaaCaaaleqabaGaaG4maaaaaaa@4E6D@

Η( τ,χ )= | χ | 1+α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4LdG0aae WaaeaacqaHepaDcaGGSaGaeq4XdmgacaGLOaGaayzkaaGaeyypa0Ja eyOeI0YaaqWaaeaacqaHhpWyaiaawEa7caGLiWoadaahaaWcbeqaai aaigdacqGHRaWkcqaHXoqyaaaaaa@474E@

γ= log 2λ ( 2μ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4SdCMaey ypa0JaciiBaiaac+gacaGGNbWaaSbaaSqaaiaaikdacqaH7oaBaeqa aOWaaeWaaeaacaaIYaGaeqiVd0gacaGLOaGaayzkaaaaaa@4215@

1 2 μ< 2 π 1 λ π MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaGOmaaaacqGHKjYOcqaH8oqBcqGH8aapcaaIYaWaaWba aSqabeaacqaHapaCdaahaaadbeqaaiabgkHiTaaaliabgkHiTiaaig daaaGccqaH7oaBdaahaaWcbeqaaiabec8aWnaaCaaameqabaGaeyOe I0caaaaaaaa@4624@

| υ | π( τ ) >1 | υ | π( τ ) π ρ( υ ) | υ | π( τ ) π + MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacq aHfpqDaiaawEa7caGLiWoadaWgaaWcbaGaeqiWda3aaeWaaeaacqaH epaDaiaawIcacaGLPaaaaeqaaOGaeyOpa4JaaGymaiabgkDiEpaaem aabaGaeqyXduhacaGLhWUaayjcSdWaa0baaSqaaiabec8aWnaabmaa baGaeqiXdqhacaGLOaGaayzkaaaabaGaeqiWda3aaWbaaWqabeaacq GHsislaaaaaOGaeyizImQaeqyWdi3aaeWaaeaacqaHfpqDaiaawIca caGLPaaacqGHKjYOdaabdaqaaiabew8a1bGaay5bSlaawIa7amaaDa aaleaacqaHapaCdaqadaqaaiabes8a0bGaayjkaiaawMcaaaqaaiab ec8aWnaaCaaameqabaGaey4kaScaaaaaaaa@66AC@

Β Τ 1,π( τ ) = Β ˜ Τ 1,π( τ ) Ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOKdi0aa0 baaSqaaiabfs6aubqaaiaaigdacaGGSaGaeqiWda3aaeWaaeaacqaH epaDaiaawIcacaGLPaaaaaGccqGH9aqpcuqHsoGqgaacamaaDaaale aacqqHKoavaeaacaaIXaGaaiilaiabec8aWnaabmaabaGaeqiXdqha caGLOaGaayzkaaaaaOGaeyyLIuSaeSyhHe6aaWbaaSqabeaacqqHDo Gtaaaaaa@4F54@

υ( χ,τ )= γ 1 ( τ )+ γ 2 ( τ )lnρ+ υ 0 ( χ,τ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyXdu3aae WaaeaacqaHhpWycaGGSaGaeqiXdqhacaGLOaGaayzkaaGaeyypa0Ja eq4SdC2aaSbaaSqaaiaaigdaaeqaaOWaaeWaaeaacqaHepaDaiaawI cacaGLPaaacqGHRaWkcqaHZoWzdaWgaaWcbaGaaGOmaaqabaGcdaqa daqaaiabes8a0bGaayjkaiaawMcaaiGacYgacaGGUbGaeqyWdiNaey 4kaSIaeqyXdu3aaSbaaSqaaiaaicdaaeqaaOWaaeWaaeaacqaHhpWy caGGSaGaeqiXdqhacaGLOaGaayzkaaaaaa@581A@

| υ 1 |Γ | χ | Im λ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacq aHfpqDdaWgaaWcbaGaaGymaaqabaaakiaawEa7caGLiWoacqGHKjYO cqqHtoWrdaabdaqaaiabeE8aJbGaay5bSlaawIa7amaaCaaaleqaba Gaciysaiaac2gacqaH7oaBdaWgaaadbaGaaGymaaqabaaaaaaa@4850@

Imλ( τ )=β+α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciysaiaac2 gacqaH7oaBdaqadaqaaiabes8a0bGaayjkaiaawMcaaiabg2da9iab ek7aIjabgUcaRiabeg7aHbaa@41E1@

Im λ Ν 0 ( τ 0 )= β 1 +2μ+ λ 1 ν 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciysaiaac2 gacqaH7oaBdaWgaaWcbaGaeuyNd40aaSbaaWqaaiaaicdaaeqaaaWc beaakmaabmaabaGaeqiXdq3aaSbaaSqaaiaaicdaaeqaaaGccaGLOa GaayzkaaGaeyypa0JaeyOeI0IaeqOSdi2aaSbaaSqaaiaaigdaaeqa aOGaey4kaSIaaGOmaiabeY7aTjabgUcaRiabeU7aSnaaBaaaleaaca aIXaaabeaakiabgkHiTmaalaaabaGaeqyVd4gabaGaaGOmaaaaaaa@4F1B@

φ ^ = κ 0 φ+ Λ 1 υ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqOXdOMbaK aacqGH9aqpcqaH6oWAdaWgaaWcbaGaaGimaaqabaGccqaHgpGAcqGH RaWkcqqHBoatdaWgaaWcbaGaaGymaaqabaGccqaHfpqDaaa@4238@

Λ Σ δ = Α δ + Ρ δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4MdWKaeu 4Odm1aaSbaaSqaaiabes7aKbqabaGccqGH9aqpcqqHroqqdaWgaaWc baGaeqiTdqgabeaakiabgUcaRiabfg6asnaaBaaaleaacqaH0oazae qaaaaa@4340@

ι ( 1 ) μ1 Λυ υ τ =φ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyUdK2aae WaaeaacqGHsislcaaIXaaacaGLOaGaayzkaaWaaWbaaSqabeaacqaH 8oqBcqGHsislcaaIXaaaaOGaeu4MdWKaeqyXduNaeyOeI0IaeqyXdu 3aaSbaaSqaaiabes8a0bqabaGccqGH9aqpcqaHgpGAaaa@491A@

χ 0 +χ+ χ 2 + χ 3 + χ 4 + χ 5 + χ 6 ++ χ ν1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaW baaSqabeaacaaIWaaaaOGaey4kaSIaeq4XdmMaey4kaSIaeq4Xdm2a aWbaaSqabeaacaaIYaaaaOGaey4kaSIaeq4Xdm2aaWbaaSqabeaaca aIZaaaaOGaey4kaSIaeq4Xdm2aaWbaaSqabeaacaaI0aaaaOGaey4k aSIaeq4Xdm2aaWbaaSqabeaacaaI1aaaaOGaey4kaSIaeq4Xdm2aaW baaSqabeaacaaI2aaaaOGaey4kaSIaeSOjGSKaey4kaSIaeq4Xdm2a aWbaaSqabeaacqaH9oGBcqGHsislcaaIXaaaaaaa@5528@

( χ+ψ+ζ ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aHhpWycqGHRaWkcqaHipqEcqGHRaWkcqaH2oGEaiaawIcacaGLPaaa daahaaWcbeqaaiaaikdaaaaaaa@3F6F@

( χ+ψ ) 2 +ζ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aHhpWycqGHRaWkcqaHipqEaiaawIcacaGLPaaadaahaaWcbeqaaiaa ikdaaaGccqGHRaWkcqaH2oGEaaa@3F79@

χ+ ( ψ+ζ ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4XdmMaey 4kaSYaaeWaaeaacqaHipqEcqGHRaWkcqaH2oGEaiaawIcacaGLPaaa daahaaWcbeqaaiaaikdaaaaaaa@3F6F@

χ+ψ+ ζ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4XdmMaey 4kaSIaeqiYdKNaey4kaSIaeqOTdO3aaWbaaSqabeaacaaIYaaaaaaa @3DE6@

χ 0 + χ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaS baaSqaaiaaicdaaeqaaOGaey4kaSIaeq4Xdm2aaSbaaSqaaiaaigda aeqaaaaa@3C1E@

χ ˙ + χ ¨ + χ ˜ + χ ^ + χ + χ + χ + χ * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4XdmMbai aacqGHRaWkcuaHhpWygaWaaiabgUcaRiqbeE8aJzaaiaGaey4kaSIa fq4XdmMbaKaacqGHRaWkcuaHhpWygaGbaiabgUcaRiqbeE8aJzaasa Gaey4kaSIafq4XdmMbauaacqGHRaWkcqaHhpWydaahaaWcbeqaaiaa cQcaaaaaaa@4B1B@

α 1,1 + α 2,2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aaW baaSqabeaacaaIXaGaaiilaiaaigdaaaGccqGHRaWkcqaHXoqydaah aaWcbeqaaiaaikdacaGGSaGaaGOmaaaaaaa@3EC9@

υ,υ φ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaaWaaeaacq aHfpqDcaGGSaGaeqyXduhacaGLPmIaayPkJaWaaSbaaSqaaiabeA8a Qbqabaaaaa@3DEE@

κ α κ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabuaeaacq aHXoqydaWgaaWcbaGaeqOUdSgabeaaaeaacqaH6oWAcqGHiiIZcqWI DesOaeqaniabggHiLdaaaa@4032@

ι=1 Ν α 2 +β+ γ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaacq aHXoqydaahaaWcbeqaaiaaikdaaaaabaGaeqyUdKMaeyypa0JaaGym aaqaaiabf25aobqdcqGHris5aOGaey4kaSIaeqOSdiMaey4kaSIaeq 4SdC2aaWbaaSqabeaacaaIYaaaaaaa@459E@

ε ε ε χ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaW baaSqabeaacqaH1oqzdaahaaadbeqaaiabew7aLnaaCaaabeqaaiab eE8aJbaaaaaaaaaa@3D20@

ε ε ε χ ε ε χ ε χ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaW baaSqabeaacqaH1oqzdaahaaadbeqaaiabew7aLnaaCaaabeqaaiab eE8aJbaaaaaaaOGaeyyXICTaeqyTdu2aaWbaaSqabeaacqaH1oqzda ahaaadbeqaaiabeE8aJbaaaaGccqGHflY1cqaH1oqzdaahaaWcbeqa aiabeE8aJbaaaaa@4AB3@

ε ( ε ε χ + ε χ +χ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaW baaSqabeaadaqadaqaaiabew7aLnaaCaaameqabaGaeqyTdu2aaWba aeqabaGaeq4XdmgaaaaaliabgUcaRiabew7aLnaaCaaameqabaGaeq 4XdmgaaSGaey4kaSIaeq4XdmgacaGLOaGaayzkaaaaaaaa@45C6@

ι=1 ν α ι =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaacq aHXoqydaWgaaWcbaGaeqyUdKgabeaaaeaacqaH5oqAcqGH9aqpcaaI XaaabaGaeqyVd4ganiabggHiLdGccqGH9aqpcaaIXaaaaa@429B@

ι=1 ν α ι + β ι =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaacq aHXoqydaWgaaWcbaGaeqyUdKgabeaakiabgUcaRiabek7aInaaBaaa leaacqaH5oqAaeqaaaqaaiabeM7aPjabg2da9iaaigdaaeaacqaH9o GBa0GaeyyeIuoakiabg2da9iaaigdaaaa@4703@

1+χ+ χ 2 + χ 3 + χ 4 ++ χ ν1 += 1 1+χ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgU caRiabeE8aJjabgUcaRiabeE8aJnaaCaaaleqabaGaaGOmaaaakiab gUcaRiabeE8aJnaaCaaaleqabaGaaG4maaaakiabgUcaRiabeE8aJn aaCaaaleqabaGaaGinaaaakiabgUcaRiablAciljabgUcaRiabeE8a JnaaCaaaleqabaGaeqyVd4MaeyOeI0IaaGymaaaakiabgUcaRiablA ciljabg2da9maalaaabaGaaGymaaqaaiaaigdacqGHRaWkcqaHhpWy aaaaaa@534F@

χ χ 2 2 + χ 3 3 χ 4 4 + χ 5 5 ±=log( 1+χ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4XdmMaey OeI0YaaSaaaeaacqaHhpWydaahaaWcbeqaaiaaikdaaaaakeaacaaI YaaaaiabgUcaRmaalaaabaGaeq4Xdm2aaWbaaSqabeaacaaIZaaaaa GcbaGaaG4maaaacqGHsisldaWcaaqaaiabeE8aJnaaCaaaleqabaGa aGinaaaaaOqaaiaaisdaaaGaey4kaSYaaSaaaeaacqaHhpWydaahaa WcbeqaaiaaiwdaaaaakeaacaaI1aaaaiabgglaXkablAciljabg2da 9iGacYgacaGGVbGaai4zamaabmaabaGaaGymaiabgUcaRiabeE8aJb GaayjkaiaawMcaaaaa@54F3@

( α+β ) 3 = α 3 +3 α 2 β+3α β 2 + β 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aHXoqycqGHRaWkcqaHYoGyaiaawIcacaGLPaaadaahaaWcbeqaaiaa iodaaaGccqGH9aqpcqaHXoqydaahaaWcbeqaaiaaiodaaaGccqGHRa WkcaaIZaGaeyyXICTaeqySde2aaWbaaSqabeaacaaIYaaaaOGaeyyX ICTaeqOSdiMaey4kaSIaaG4maiabgwSixlabeg7aHjabgwSixlabek 7aInaaCaaaleqabaGaaGOmaaaakiabgUcaRiabek7aInaaCaaaleqa baGaaG4maaaaaaa@5868@

χ 1 κ + χ 2 κ + χ 3 κ ++ χ ν κ =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aa0 baaSqaaiaaigdaaeaacqaH6oWAaaGccqGHRaWkcqaHhpWydaqhaaWc baGaaGOmaaqaaiabeQ7aRbaakiabgUcaRiabeE8aJnaaDaaaleaaca aIZaaabaGaeqOUdSgaaOGaey4kaSIaeSOjGSKaey4kaSIaeq4Xdm2a a0baaSqaaiabe27aUbqaaiabeQ7aRbaakiabg2da9iaaicdaaaa@4ECD@

χ κ 1 + χ κ 2 + χ κ 3 + χ κ ν =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaS baaSqaaiabeQ7aRnaaCaaameqabaGaaGymaaaaaSqabaGccqGHRaWk cqaHhpWydaWgaaWcbaGaeqOUdS2aaWbaaWqabeaacaaIYaaaaaWcbe aakiabgUcaRiabeE8aJnaaBaaaleaacqaH6oWAdaahaaadbeqaaiaa iodaaaaaleqaaOGaey4kaSIaeSOjGSKaeq4Xdm2aaSbaaSqaaiabeQ 7aRnaaCaaameqabaGaeqyVd4gaaaWcbeaakiabg2da9iaaicdaaaa@4ECB@

χ κ 1 + χ κ 2 + χ κ 3 ++ χ κ ν =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaS baaSqaaiabeQ7aRnaaBaaameaacaaIXaaabeaaaSqabaGccqGHRaWk cqaHhpWydaWgaaWcbaGaeqOUdS2aaSbaaWqaaiaaikdaaeqaaaWcbe aakiabgUcaRiabeE8aJnaaBaaaleaacqaH6oWAdaWgaaadbaGaaG4m aaqabaaaleqaaOGaey4kaSIaeSOjGSKaey4kaSIaeq4Xdm2aaSbaaS qaaiabeQ7aRnaaBaaameaacqaH9oGBaeqaaaWcbeaakiabg2da9iaa icdaaaa@4FA9@

χ 2 ψ χ 2 ψ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaW baaSqabeaacaaIYaWaaWbaaWqabeaacqaHipqEaaaaaOGaeyiyIKRa eq4Xdm2aaWbaaSqabeaacaaIYaWaaWbaaWqabeaacqaHipqEaaaaaa aa@4100@

α χ 2 +βχ+γ=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey yXICTaeq4Xdm2aaWbaaSqabeaacaaIYaaaaOGaey4kaSIaeqOSdiMa eyyXICTaeq4XdmMaey4kaSIaeq4SdCMaeyypa0JaaGimaaaa@4757@

log3 (χ+ψ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac+ gacaGGNbGaaG4maiaacIcacqaHhpWycqGHRaWkcqaHipqEcaGGPaWa aWbaaSqabeaacaaIZaaaaaaa@402E@

log 2 χ2logχ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac+ gacaGGNbWaaWbaaSqabeaacaaIYaaaaOGaeq4XdmMaeyiyIKRaaGOm aiabgwSixlGacYgacaGGVbGaai4zaiabeE8aJbaa@44C5@

logχ logα = log α χ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaci GGSbGaai4BaiaacEgacqaHhpWyaeaaciGGSbGaai4BaiaacEgacqaH XoqyaaGaeyypa0JaciiBaiaac+gacaGGNbWaaSbaaSqaaiabeg7aHb qabaGccqaHhpWyaaa@465F@

χ ¯ +χ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafq4XdmMbae bacqGHRaWkcqaHhpWyaaa@3A5F@

arg( ζ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyyaiaack hacaGGNbWaaeWaaeaacqaH2oGEaiaawIcacaGLPaaaaaa@3C05@

 

4.2 Complex

υ ( 0 ) = ζ=1 ν ξ ζ ( 1 ) ρ ζ ( υ ( ι ) ) = ζ=1 ν ξ ζ ( 2 ) ρ ζ ( υ ( 2 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyXdu3aaW baaSqabeaadaqadaqaaiaaicdaaiaawIcacaGLPaaaaaGccqGH9aqp daaeWbqaaiabe67a4naaBaaaleaacqaH2oGEaeqaaOWaaWbaaSqabe aadaqadaqaaiaaigdaaiaawIcacaGLPaaaaaGccqaHbpGCdaWgaaWc baGaeqOTdOhabeaakmaabmaabaGaeqyXdu3aaWbaaSqabeaadaqada qaaiabeM7aPbGaayjkaiaawMcaaaaaaOGaayjkaiaawMcaaaWcbaGa eqOTdONaeyypa0JaaGymaaqaaiabe27aUbqdcqGHris5aOGaeyypa0 ZaaabCaeaacqaH+oaEdaWgaaWcbaGaeqOTdOhabeaakmaaCaaaleqa baWaaeWaaeaacaaIYaaacaGLOaGaayzkaaaaaOGaeqyWdi3aaSbaaS qaaiabeA7a6bqabaGcdaqadaqaaiabew8a1naaCaaaleqabaWaaeWa aeaacaaIYaaacaGLOaGaayzkaaaaaaGccaGLOaGaayzkaaaaleaacq aH2oGEcqGH9aqpcaaIXaaabaGaeqyVd4ganiabggHiLdaaaa@6B92@

| υ | = ορισμος ( κ=1 ν υ κ 2 ) 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaqWaaeaacq aHfpqDaiaawEa7caGLiWoadaWfGaqaaiabg2da9aWcbeqaaiabe+7a Vjabeg8aYjabeM7aPjabeo8aZjabeY7aTjabe+7aVjabek8awbaakm aabmaabaWaaabCaeaacqaHfpqDdaqhaaWcbaGaeqOUdSgabaGaaGOm aaaaaeaacqaH6oWAcqGH9aqpcaaIXaaabaGaeqyVd4ganiabggHiLd aakiaawIcacaGLPaaadaahaaWcbeqaamaalaaabaGaaGymaaqaaiaa ikdaaaaaaaaa@5747@

0 | ξ ι ( 1 ) ( τ,χ ) |dχ Γ( φ ( 0 ) Λ 1 ( + ) + α ( 0 ) Λ 1 ( + ) + η ( 0 ) Λ 1 ( + ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWdXb qaamaaemaabaGaeqOVdG3aaSbaaSqaaiabeM7aPbqabaGcdaahaaWc beqaamaabmaabaGaaGymaaGaayjkaiaawMcaaaaakmaabmaabaGaeq iXdqNaaiilaiabeE8aJbGaayjkaiaawMcaaaGaay5bSlaawIa7aiaa dsgacqaHhpWyaSqaaiaaicdaaeaacqGHEisPa0Gaey4kIipakiabgs MiJcqaaiabfo5ahnaabmaabaWaauWaaeaacqaHgpGAdaahaaWcbeqa amaabmaabaGaaGimaaGaayjkaiaawMcaaaaaaOGaayzcSlaawQa7am aaBaaaleaacqqHBoatdaahaaadbeqaaiaaigdaaaWcdaqadaqaaiab l2riHoaaCaaameqabaGaey4kaScaaaWccaGLOaGaayzkaaaabeaaki abgUcaRmaafmaabaGaeqySde2aaWbaaSqabeaadaqadaqaaiaaicda aiaawIcacaGLPaaaaaaakiaawMa7caGLkWoadaWgaaWcbaGaeu4MdW 0aaWbaaWqabeaacaaIXaaaaSWaaeWaaeaacqWIDesOdaahaaadbeqa aiabgUcaRaaaaSGaayjkaiaawMcaaaqabaGccqGHRaWkdaqbdaqaai abeE7aOnaaCaaaleqabaWaaeWaaeaacaaIWaaacaGLOaGaayzkaaaa aaGccaGLjWUaayPcSdWaaSbaaSqaaiabfU5amnaaCaaameqabaGaaG ymaaaalmaabmaabaGaeSyhHe6aaWbaaWqabeaacqGHRaWkaaaaliaa wIcacaGLPaaaaeqaaaGccaGLOaGaayzkaaaaaaa@7CA4@

Β ˜ 1 ( Τ )= max ι=1,ν max ξι sup Γ ξ Γ ξ | υ ι |dτ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafuOKdiKbaG aadaWgaaWcbaGaaGymaaqabaGcdaqadaqaaiabfs6aubGaayjkaiaa wMcaaiabg2da9maaxababaGaciyBaiaacggacaGG4baaleaacqaH5o qAcqGH9aqpcaaIXaGaaiilaiablAciljabe27aUbqabaGcdaWfqaqa aiGac2gacaGGHbGaaiiEaaWcbaGaeqOVdGNaeyiyIKRaeqyUdKgabe aakmaaxababaGaci4CaiaacwhacaGGWbaaleaacqqHtoWrdaWgaaad baGaeqOVdGhabeaaaSqabaGcdaWdrbqaamaaemaabaGaeqyXdu3aaS baaSqaaiabeM7aPbqabaaakiaawEa7caGLiWoacaWGKbGaeqiXdqha leaacqqHtoWrdaWgaaadbaGaeqOVdGhabeaaaSqab0Gaey4kIipaaa a@6484@

{ ε λ } λ=1 ν+μ2κ ={ { ε ι } ιΑ , { ε ξ } ξ Α } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacq aH1oqzdaWgaaWcbaGaeq4UdWgabeaakmaaCaaaleqabaGccWaGGBOm Gi6ccaaMb8UccWaGGBOmGikaaaGaay5Eaiaaw2haamaaDaaaleaacq aH7oaBcqGH9aqpcaaIXaaabaGaeqyVd4Maey4kaSIaeqiVd0MaeyOe I0IaaGOmaiabeQ7aRbaakiabg2da9maacmaabaWaaiWaaeaacqaH1o qzdaWgaaWcbaGaeqyUdKgabeaaaOGaay5Eaiaaw2haamaaBaaaleaa cqaH5oqAcqGHjiYZcqqHroqqaeqaaOGaaiilamaacmaabaGaeqyTdu 2aaSbaaSqaaiabe67a4bqabaGcdaahaaWcbeqaaOGamai4gkdiIcaa aiaawUhacaGL9baadaWgaaWcbaGaeqOVdGNaeyycI8SafuyKdeKbau aaaeqaaaGccaGL7bGaayzFaaaaaa@6B00@

υ ε (ν+1,μ+1) υ+ κ=1 ν ψ κ υ κ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaey4bIeTaeq yXdu3aaSbaaSqaaiabew7aLbqabaGcdaGdKaWcbaGaaiikaiabe27a UjabgUcaRiaaigdacaGGSaGaeqiVd0Maey4kaSIaaGymaiaacMcaae qakiaawkziaiabgEGirlabew8a1jabgUcaRmaaqahabaGaey4bIe9a aSbaaSqaaiabeI8a5naaBaaameaacqaH6oWAaeqaaaWcbeaakiabew 8a1naaBaaaleaacqaH6oWAaeqaaaqaaiabeQ7aRjabg2da9iaaigda aeaacqaH9oGBa0GaeyyeIuoaaaa@5A14@

wsep,1 μκ ={ ( ε, { ε ξ } ξ=1 μ ) wsep μκ : ε 2 ε μ 0 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyyeHe8aa0 baaSqaaiaadEhacaWGZbGaamyzaiaadchacaGGSaGaaGymaaqaaiab eY7aTjablYJi6iabeQ7aRbaakiabg2da9maacmaabaWaaeWaaeaacq aH1oqzcaGGSaWaaiWaaeaacuaH1oqzgaqbamaaBaaaleaacqaH+oaE aeqaaaGccaGL7bGaayzFaaWaa0baaSqaaiabe67a4jabg2da9iaaig daaeaacqaH8oqBaaaakiaawIcacaGLPaaacqGHiiIZcqGHresWdaqh aaWcbaGaam4DaiaadohacaWGLbGaamiCaaqaaiabeY7aTjablYJi6i abeQ7aRbaakiaacQdadaWcaaqaaiabew7aLnaaCaaaleqabaGaaGOm aaaaaOqaaiabew7aLnaaBaaaleaacqaH8oqBaeqaaOWaaWbaaSqabe aakiadacUHYaIOaaaaaiabgkziUkaaicdaaiaawUhacaGL9baaaaa@6C40@

ε 1 ={ ε,{ ε 0,2 , ε 0,5 ,ε, ε 1,2 , ε 1,5 | logε | } } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaaigdaaeqaaOGaeyypa0ZaaiWaaeaacqaH1oqzcaGGSaWa aiWaaeaacqaH1oqzdaahaaWcbeqaaiaaicdacaGGSaGaaGOmaaaaki aacYcacqaH1oqzdaahaaWcbeqaaiaaicdacaGGSaGaaGynaaaakiaa cYcacqaH1oqzcaGGSaGaeqyTdu2aaWbaaSqabeaacaaIXaGaaiilai aaikdaaaGccaGGSaWaaSaaaeaacqaH1oqzdaahaaWcbeqaaiaaigda caGGSaGaaGynaaaaaOqaamaaemaabaGaciiBaiaac+gacaGGNbGaeq yTdugacaGLhWUaayjcSdaaaaGaay5Eaiaaw2haaaGaay5Eaiaaw2ha aaaa@5C76@

υ μ ν + ν μ ν +κ d υ μ ν dχ =κ× φ μ ν + υ μ ν1 + ν μ ν1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyXdu3aa0 baaSqaaiabeY7aTbqaaiabe27aUbaakiabgUcaRiabe27aUnaaDaaa leaacqaH8oqBaeaacqaH9oGBaaGccqGHRaWkcqaH6oWAdaWcaaqaai aadsgacqaHfpqDdaqhaaWcbaGaeqiVd0gabaGaeqyVd4gaaaGcbaGa amizaiabeE8aJbaacqGH9aqpcqaH6oWAcqGHxdaTcqaHgpGAdaqhaa WcbaGaeqiVd0gabaGaeqyVd4gaaOGaey4kaSIaeqyXdu3aa0baaSqa aiabeY7aTbqaaiabe27aUjabgkHiTiaaigdaaaGccqGHRaWkcqaH9o GBdaqhaaWcbaGaeqiVd0gabaGaeqyVd4MaeyOeI0IaaGymaaaaaaa@673F@

Ω | κ μ ξ | 2 dχ+κ ν=1 ξ | κ μ ν | 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8quaeaada abdaqaaiabeQ7aRnaaDaaaleaacqaH8oqBaeaacqaH+oaEaaaakiaa wEa7caGLiWoadaahaaWcbeqaaiaaikdaaaGccaWGKbGaeq4XdmMaey 4kaSIaeqOUdS2aaabCaeaadaabdaqaaiabeQ7aRnaaDaaaleaacqaH 8oqBaeaacqaH9oGBaaaakiaawEa7caGLiWoadaahaaWcbeqaaiaaik daaaaabaGaeqyVd4Maeyypa0JaaGymaaqaaiabe67a4bqdcqGHris5 aaWcbaGaeuyQdCfabeqdcqGHRiI8aaaa@594F@

sgn ξΙ( δ 1 , δ 2 ) α ξ Γ ι 1 ε ξ dχ =sgn ξΙ( δ 1 , δ 2 ) α ξ Γ ι 2 ε ξ dχ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacE gacaGGUbWaaabuaeaacqaHXoqydaWgaaWcbaGaeqOVdGhabeaakmaa pefabaGaeqyTdu2aaSbaaSqaaiabe67a4bqabaGccaWGKbGaeq4Xdm galeaacqqHtoWrdaWgaaadbaGaeqyUdK2aaSbaaeaacaaIXaaabeaa aeqaaaWcbeqdcqGHRiI8aaWcbaGaeqOVdGNaeyicI4SaeuyMdK0aae WaaeaacqaH0oazdaWgaaadbaGaaGymaaqabaWccaGGSaGaeqiTdq2a aSbaaWqaaiaaikdaaeqaaaWccaGLOaGaayzkaaaabeqdcqGHris5aO Gaeyypa0JaeyOeI0Iaci4CaiaacEgacaGGUbWaaabuaeaacqaHXoqy daWgaaWcbaGaeqOVdGhabeaakmaapefabaGaeqyTdu2aaSbaaSqaai abe67a4bqabaGccaWGKbGaeq4XdmgaleaacqqHtoWrdaWgaaadbaGa eqyUdK2aaSbaaeaacaaIYaaabeaaaeqaaaWcbeqdcqGHRiI8aaWcba GaeqOVdGNaeyicI4SaeuyMdK0aaeWaaeaacqaH0oazdaWgaaadbaGa aGymaaqabaWccaGGSaGaeqiTdq2aaSbaaWqaaiaaikdaaeqaaaWcca GLOaGaayzkaaaabeqdcqGHris5aaaa@7AC3@

φ( Φ σ κ )= φ( Φ σ κ ) φ κ ( Φ σ κ ) 0 + φ κ ( Φ σ κ ) χ κ χ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdO2aae WaaeaacqqHMoGrdaqhaaWcbaGaeq4WdmhabaGaeqOUdSgaaaGccaGL OaGaayzkaaGaeyypa0ZaaGbaaeaacqaHgpGAdaqadaqaaiabfA6agn aaDaaaleaacqaHdpWCaeaacqaH6oWAaaaakiaawIcacaGLPaaacqGH sislcqaHgpGAdaWgaaWcbaGaeqOUdSgabeaakmaabmaabaGaeuOPdy 0aa0baaSqaaiabeo8aZbqaaiabeQ7aRbaaaOGaayjkaiaawMcaaaWc baWaa4akaWqabeaacqGHsgIRaSGaayPKHaGaaGimaaGccaGL44pacq GHRaWkdaagaaqaaiabeA8aQnaaBaaaleaacqaH6oWAaeqaaOWaaeWa aeaacqqHMoGrdaqhaaWcbaGaeq4WdmhabaGaeqOUdSgaaaGccaGLOa GaayzkaaaaleaacqaHhpWydaahaaadbeqaaiabeQ7aRbaaaOGaayjo +dGaeyOKH4Qaeq4Xdmgaaa@6DCC@

φ(χ)=φ(α)+φ'(α)×(χα)++ φ ν (α) ν × (χα) ν + φ (ν+1) ( γ ) ν+1 × ( χα ) (ν+1) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdOMaai ikaiabeE8aJjaacMcacqGH9aqpcqaHgpGAcaGGOaGaeqySdeMaaiyk aiabgUcaRiabeA8aQjaacEcacaGGOaGaeqySdeMaaiykaiabgEna0k aacIcacqaHhpWycqGHsislcqaHXoqycaGGPaGaey4kaSIaeSOjGSKa ey4kaSYaaSaaaeaacqaHgpGAdaahaaWcbeqaaiabe27aUbaakiaacI cacqaHXoqycaGGPaaabaGaeqyVd4gaaiabgEna0kaacIcacqaHhpWy cqGHsislcqaHXoqycaGGPaWaaWbaaSqabeaacqaH9oGBaaGccqGHRa WkdaWcaaqaaiabeA8aQnaaCaaaleqabaGaaiikaiabe27aUjabgUca RiaaigdacaGGPaaaaOWaaeWaaeaacqaHZoWzaiaawIcacaGLPaaaae aacqaH9oGBcqGHRaWkcaaIXaaaaiabgEna0oaabmaabaGaeq4XdmMa eyOeI0IaeqySdegacaGLOaGaayzkaaWaaWbaaSqabeaacaGGOaGaeq yVd4Maey4kaSIaaGymaiaacMcaaaaaaa@7E3F@

2 υ χψ =( χ 2 ψ 2 )×{ τ×φ"(τ)+3×φ'(τ) } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHciITdaahaaWcbeqaaiaaikdaaaGccqaHfpqDaeaacqGHciITcqaH hpWycqGHciITcqaHipqEaaGaeyypa0ZaaeWaaeaacqaHhpWydaahaa WcbeqaaiaaikdaaaGccqGHsislcqaHipqEdaahaaWcbeqaaiaaikda aaaakiaawIcacaGLPaaacqGHxdaTdaGadaqaaiabes8a0jabgEna0k abeA8aQjaackcacaGGOaGaeqiXdqNaaiykaiabgUcaRiaaiodacqGH xdaTcqaHgpGAcaGGNaGaaiikaiabes8a0jaacMcaaiaawUhacaGL9b aaaaa@6040@

log( 1+χ )log( 1χ )=log 1+χ 1χ = ι=1 χ 2ι1 2ι1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiBaiaac+ gacaGGNbWaaeWaaeaacaaIXaGaey4kaSIaeq4XdmgacaGLOaGaayzk aaGaeyOeI0IaciiBaiaac+gacaGGNbWaaeWaaeaacaaIXaGaeyOeI0 Iaeq4XdmgacaGLOaGaayzkaaGaeyypa0JaciiBaiaac+gacaGGNbWa aSaaaeaacaaIXaGaey4kaSIaeq4XdmgabaGaaGymaiabgkHiTiabeE 8aJbaacqGH9aqpdaaeWbqaamaalaaabaGaeq4Xdm2aaWbaaSqabeaa caaIYaGaeyyXICTaeqyUdKMaeyOeI0IaaGymaaaaaOqaaiaaikdacq GHflY1cqaH5oqAcqGHsislcaaIXaaaaaWcbaGaeqyUdKMaeyypa0Ja aGymaaqaaiabg6HiLcqdcqGHris5aaaa@67C3@

Δ χ ν ω= 0λν κ 1 ++ κ ν =λ κ 1 +2 κ 2 ++ν κ ν =ν κ 1 ,, κ ν 0 Δ υ λ ω× n! ( Δ χ 1 υ ) κ 1 ( Δ χ ν υ ) κ ν κ 1 ! (1!) κ 1 κ ν ! ( ν! ) κ ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdq0aa0 baaSqaaiabeE8aJbqaaiabe27aUbaakiabeM8a3jabg2da9maaqaba baWaaabeaeaacqqHuoardaqhaaWcbaGaeqyXduhabaGaeq4UdWgaaO GaeqyYdCNaey41aq7aaSaaaeaacaWGUbGaaiyiamaabmaabaGaeuiL dq0aa0baaSqaaiabeE8aJbqaaiaaigdaaaGccqaHfpqDaiaawIcaca GLPaaadaahaaWcbeqaaiabeQ7aRnaaCaaameqabaGaaGymaaaaaaGc cqWIMaYsdaqadaqaaiabfs5aenaaDaaaleaacqaHhpWyaeaacqaH9o GBaaGccqaHfpqDaiaawIcacaGLPaaadaahaaWcbeqaaiabeQ7aRnaa BaaameaacqaH9oGBaeqaaaaaaOqaaiabeQ7aRnaaBaaaleaacaaIXa aabeaakiaacgcacaGGOaGaaGymaiaacgcacaGGPaWaaWbaaSqabeaa cqaH6oWAdaWgaaadbaGaaGymaaqabaaaaOGaeSOjGSKaeqOUdS2aaS baaSqaaiabe27aUbqabaGccaGGHaWaaeWaaeaacqaH9oGBcaGGHaaa caGLOaGaayzkaaWaaWbaaSqabeaacqaH6oWAdaWgaaadbaGaeqyVd4 gabeaaaaaaaaWceaqabeaacqaH6oWAdaWgaaadbaGaaGymaaqabaWc cqGHRaWkcqWIMaYscqGHRaWkcqaH6oWAdaWgaaadbaGaeqyVd4gabe aaliabg2da9iabeU7aSbqaaiabeQ7aRnaaBaaameaacaaIXaaabeaa liabgUcaRiaaikdacqaH6oWAdaWgaaadbaGaaGOmaaqabaWccqGHRa WkcqWIMaYscqGHRaWkcqaH9oGBcqaH6oWAdaWgaaadbaGaeqyVd4ga beaaliabg2da9iabe27aUbqaaiabeQ7aRnaaBaaameaacaaIXaaabe aaliaacYcacqWIMaYscaGGSaGaeqOUdS2aaSbaaWqaaiabe27aUbqa baWccqGHLjYScaaIWaaaaeqaniabggHiLdaaleaacaaIWaGaeyizIm Qaeq4UdWMaeyizImQaeqyVd4gabeqdcqGHris5aaaa@AA4F@

1ιν α ι =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabeaeaacq aHXoqydaWgaaWcbaGaeqyUdKgabeaaaeaacaaIXaGaeyizImQaeqyU dKMaeyizImQaeqyVd4gabeqdcqGHris5aOGaeyypa0JaaGymaaaa@44A0@

κ=1 ( 1 ) κ+1 ( κ+1 )×ln(κ+1) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaada WcaaqaamaabmaabaGaeyOeI0IaaGymaaGaayjkaiaawMcaamaaCaaa leqabaGaeqOUdSMaey4kaSIaaGymaaaaaOqaamaabmaabaGaeqOUdS Maey4kaSIaaGymaaGaayjkaiaawMcaaiabgEna0kGacYgacaGGUbGa aiikaiabeQ7aRjabgUcaRiaaigdacaGGPaaaaaWcbaGaeqOUdSMaey ypa0JaaGymaaqaaiabg6HiLcqdcqGHris5aaaa@515F@

κ=11 28 (κ10)×sin[ π κ10 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaaca GGOaGaeqOUdSMaeyOeI0IaaGymaiaaicdacaGGPaGaey41aqRaci4C aiaacMgacaGGUbWaamWaaeaadaWcaaqaaiabec8aWbqaaiabeQ7aRj abgkHiTiaaigdacaaIWaaaaaGaay5waiaaw2faaaWcbaGaeqOUdSMa eyypa0JaaGymaiaaigdaaeaacaaIYaGaaGioaaqdcqGHris5aaaa@5014@

 

 

5. Integrals

5.1 Simple

0 Τ Φ 2 ( τ,χ )dτ Δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaacq qHMoGrdaWgaaWcbaGaaGOmaaqabaGcdaqadaqaaiabes8a0jaacYca cqaHhpWyaiaawIcacaGLPaaacaWGKbGaeqiXdqhaleaacaaIWaaaba GaeuiPdqfaniabgUIiYdGccqGHLjYScqqHuoaraaa@48A5@

0 Τ η( τ )dτ =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaacq aH3oaAdaqadaqaaiabes8a0bGaayjkaiaawMcaaiaadsgacqaHepaD aSqaaiaaicdaaeaacqqHKoava0Gaey4kIipakiabg2da9iaaicdaaa a@4412@

υ Δ,υ 0 Φ = ΚΜ υ Κ Κ Δ υ 0 ( χ )dχ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaaWaaeaacq aHfpqDdaqhaaWcbaGaeuiLdqKaaiilaiabew8a1bqaaiaaicdaaaaa kiaawMYicaGLQmcadaWgaaWcbaGaeuOPdyeabeaakiabg2da9iabgk HiTmaaqafabaGaeqyXdu3aaSbaaSqaaiabfQ5albqabaGcdaWdrbqa aiabgs5aejabew8a1naaCaaaleqabaGaaGimaaaakmaabmaabaGaeq 4XdmgacaGLOaGaayzkaaGaamizaiabeE8aJbWcbaGaeuOMdSeabeqd cqGHRiI8aaWcbaGaeuOMdSKaeyicI4SaeuiNd0eabeqdcqGHris5aa aa@59BC@

υ τ ,υ + Ω υ( χ,τ )×υ( χ )dχ = Ω φ( χ,τ )υ( χ )dχ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaaada qaaiabew8a1naaBaaaleaacqaHepaDaeqaaOGaaiilaiabew8a1bGa ayzkJiaawQYiaiabgUcaRmaapefabaGaey4bIeTaeqyXdu3aaeWaae aacqaHhpWycaGGSaGaeqiXdqhacaGLOaGaayzkaaGaey41aqRaey4b IeTaeqyXdu3aaeWaaeaacqaHhpWyaiaawIcacaGLPaaacaWGKbGaeq 4XdmgaleaacqGHPoWvaeqaniabgUIiYdaakeaacqGH9aqpdaWdrbqa aiabeA8aQnaabmaabaGaeq4XdmMaaiilaiabes8a0bGaayjkaiaawM caaiabew8a1naabmaabaGaeq4XdmgacaGLOaGaayzkaaGaamizaiab eE8aJbWcbaGaeyyQdCfabeqdcqGHRiI8aaaaaa@6B09@

α( β,υ )= Ω β( χ )υ( χ )dχ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySde2aae WaaeaacqaHYoGycaGGSaGaeqyXduhacaGLOaGaayzkaaGaeyypa0Za a8quaeaacqGHhis0cqaHYoGydaqadaqaaiabeE8aJbGaayjkaiaawM caaiabgwSixlabgEGirlabew8a1naabmaabaGaeq4XdmgacaGLOaGa ayzkaaGaamizaiabeE8aJbWcbaGaeuyQdCfabeqdcqGHRiI8aaaa@53F3@

Ω φ( χ,τ )υ( χ )dχ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8quaeaacq aHgpGAdaqadaqaaiabeE8aJjaacYcacqaHepaDaiaawIcacaGLPaaa cqaHfpqDdaqadaqaaiabeE8aJbGaayjkaiaawMcaaiaadsgacqaHhp WyaSqaaiabfM6axbqab0Gaey4kIipaaaa@48E8@

Κ ( υ( χ, τ ν+1 )υ( χ, τ ν ) )dχ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8quaeaada qadaqaaiabew8a1naabmaabaGaeq4XdmMaaiilaiabes8a0naaBaaa leaacqaH9oGBcqGHRaWkcaaIXaaabeaaaOGaayjkaiaawMcaaiabgk HiTiabew8a1naabmaabaGaeq4XdmMaaiilaiabes8a0naaBaaaleaa cqaH9oGBaeqaaaGccaGLOaGaayzkaaaacaGLOaGaayzkaaGaamizai abeE8aJbWcbaGaeuOMdSeabeqdcqGHRiI8aaaa@533B@

Κ Δ υ 0 ( χ )dχ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0Yaa8 quaeaacqGHuoarcqaHfpqDdaahaaWcbeqaaiaaicdaaaGcdaqadaqa aiabeE8aJbGaayjkaiaawMcaaiaadsgacqaHhpWyaSqaaiabfQ5alb qab0Gaey4kIipaaaa@44A0@

Λ Κ, σ = Κ, σ Ιdχ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4MdW0aaS baaSqaaiabfQ5aljaacYcacuaHdpWCgaGbaaqabaGccqGH9aqpdaWd rbqaaiabfM5ajjaadsgacqaHhpWyaSqaaiabfQ5aljaacYcacuaHdp WCgaGbaaqab0Gaey4kIipaaaa@46E8@

τ ν τ ν+1 Δυ( χ,τ )dτ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaacq GHuoarcqaHfpqDdaqadaqaaiabeE8aJjaacYcacqaHepaDaiaawIca caGLPaaacaWGKbGaeqiXdqhaleaacqaHepaDdaWgaaadbaGaeqyVd4 gabeaaaSqaaiabes8a0naaBaaameaacqaH9oGBcqGHRaWkcaaIXaaa beaaa0Gaey4kIipaaaa@4CED@

0 Τ | υ( τ ) | 2 dτ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaada abdaqaaiabew8a1naabmaabaGaeqiXdqhacaGLOaGaayzkaaaacaGL hWUaayjcSdWaaWbaaSqabeaacaaIYaaaaOGaamizaiabes8a0bWcba GaaGimaaqaaiabfs6aubqdcqGHRiI8aaaa@4678@

0 Τ | υ ˙ ( τ ) | 2 dτ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaada abdaqaaiqbew8a1zaacaWaaeWaaeaacqaHepaDaiaawIcacaGLPaaa aiaawEa7caGLiWoadaahaaWcbeqaaiaaikdaaaGccaWGKbGaeqiXdq haleaacaaIWaaabaGaeuiPdqfaniabgUIiYdaaaa@4681@

0 Τ υ φdτ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaacu aHfpqDgaqbaiabeA8aQjaadsgacqaHepaDaSqaaiaaicdaaeaacqqH Koava0Gaey4kIipaaaa@40DE@

0 Τ | υ ˜ ( τ ) | π dτ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaada abdaqaaiqbew8a1zaaiaWaaeWaaeaacqaHepaDaiaawIcacaGLPaaa aiaawEa7caGLiWoadaahaaWcbeqaaiabec8aWnaaCaaameqabaGaey OeI0caaaaakiaadsgacqaHepaDaSqaaiaaicdaaeaacqqHKoava0Ga ey4kIipaaaa@48A3@

0 Τ Φ( τ,χ )dτ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaacq qHMoGrdaqadaqaaiabes8a0jaacYcacqaHhpWyaiaawIcacaGLPaaa caWGKbGaeqiXdqhaleaacaaIWaaabaGaeuiPdqfaniabgUIiYdGccq GHsgIRcqGHEisPaaa@47E5@

0 Τ 0 1 γ( τ )| υ( τ ) |dσ dτ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaada WdXbqaaiabeo7aNnaabmaabaGaeqiXdqhacaGLOaGaayzkaaWaaqWa aeaacqaHfpqDdaqadaqaaiabes8a0bGaayjkaiaawMcaaaGaay5bSl aawIa7aiaadsgacqaHdpWCaSqaaiaaicdaaeaacaaIXaaaniabgUIi YdGccaWGKbGaeqiXdqhaleaacaaIWaaabaGaeuiPdqfaniabgUIiYd aaaa@510E@

1 μ 0 Τ Φ 1 ( τ,λ υ ¯ )dτ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaeqiVd0gaamaapehabaGaeuOPdy0aaSbaaSqaaiaaigda aeqaaOWaaeWaaeaacqaHepaDcaGGSaGaeq4UdWMafqyXduNbaebaai aawIcacaGLPaaacaWGKbGaeqiXdqhaleaacaaIWaaabaGaeuiPdqfa niabgUIiYdaaaa@49CB@

Ω ρ 2 dχ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8quaeaacq aHbpGCdaahaaWcbeqaaiabgkHiTiaaikdaaaGccaWGKbGaeq4Xdmga leaacqqHPoWvaeqaniabgUIiYdaaaa@400F@

1 6 σ 1 σ( 3 σ 2 )η( σ )α( σ )dσ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaGOnaaaadaWdXbqaaiabeo8aZnaabmaabaGaaG4maiab gkHiTiabeo8aZnaaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaai abeE7aOnaabmaabaGaeq4WdmhacaGLOaGaayzkaaGaeqySde2aaeWa aeaacqaHdpWCaiaawIcacaGLPaaacaWGKbGaeq4WdmhaleaacqaHdp WCaeaacaaIXaaaniabgUIiYdaaaa@50A4@

λ max 0τ1 0 1 Γ( τ,σ )η( σ )ψ( σ,β )dσ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdW2aaC beaeaaciGGTbGaaiyyaiaacIhaaSqaaiaaicdacqGHKjYOcqaHepaD cqGHKjYOcaaIXaaabeaakmaapehabaGaeu4KdC0aaeWaaeaacqaHep aDcaGGSaGaeq4WdmhacaGLOaGaayzkaaGaeq4TdG2aaeWaaeaacqaH dpWCaiaawIcacaGLPaaacqaHipqEdaqadaqaaiabeo8aZjaacYcacq aHYoGyaiaawIcacaGLPaaacaWGKbGaeq4WdmhaleaacaaIWaaabaGa aGymaaqdcqGHRiI8aaaa@5B7B@

1 2 λ sup υΥ( ρ ) 0 1 | υ( σ ) |dσ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaaGOmaaaacqaH7oaBdaWfqaqaaiGacohacaGG1bGaaiiC aaWcbaGaeqyXduNaeyicI4SaeuyPdu1aaeWaaeaacqaHbpGCaiaawI cacaGLPaaaaeqaaOWaa8qCaeaadaabdaqaaiabew8a1naabmaabaGa eq4WdmhacaGLOaGaayzkaaaacaGLhWUaayjcSdGaamizaiabeo8aZb WcbaGaaGimaaqaaiaaigdaa0Gaey4kIipaaaa@5336@

0 1 τ Ξ( τ,σ )υ( σ )dσ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaada WcaaqaaiabgkGi2cqaaiabgkGi2kabes8a0baacqqHEoawdaqadaqa aiabes8a0jaacYcacqaHdpWCaiaawIcacaGLPaaacqaHfpqDdaqada qaaiabeo8aZbGaayjkaiaawMcaaiaadsgacqaHdpWCaSqaaiaaicda aeaacaaIXaaaniabgUIiYdaaaa@4D7A@

τ 1 υ σdσ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaada qbdaqaaiqbew8a1zaagaaacaGLjWUaayPcSdGaeq4WdmNaamizaiab eo8aZbWcbaGaeqiXdqhabaGaaGymaaqdcqGHRiI8aaaa@444A@

Η η ( χ+2 )dχ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8quaeaacq aH3oaAaSqaaiabfE5aibqab0Gaey4kIipakmaabmaabaGaeq4XdmMa ey4kaSIaaGOmaaGaayjkaiaawMcaaiaadsgacqaHhpWyaaa@42E2@

1 ε χ 2 χ1 dχ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qmaeaacq aH1oqzdaahaaWcbeqaaiabeE8aJnaaCaaameqabaGaaGOmaaaaliab gkHiTiabeE8aJjabgkHiTiaaigdaaaGccaWGKbGaeq4Xdmgaleaaca aIXaaabaGaeyOhIukaniabgUIiYdaaaa@45C2@

dχ χ =logχ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qaaeaada WcaaqaaiaadsgacqaHhpWyaeaacqaHhpWyaaaaleqabeqdcqGHRiI8 aOGaeyypa0JaciiBaiaac+gacaGGNbGaeq4Xdmgaaa@41F0@

 

 

5.2 Complex

τ ν τ ν+1 Κ υ τ ( χ,τ )dχ dτ σ Ε Κ τ ν τ ν+1 σ υ( χ,τ )× ν Κ,σ ( χ ) δ γ ( χ )dτ = τ ν τ ν+1 Κ φ( χ,τ )dχ dτ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWdXb qaamaapefabaGaeqyXdu3aaSbaaSqaaiabes8a0bqabaGcdaqadaqa aiabeE8aJjaacYcacqaHepaDaiaawIcacaGLPaaacaWGKbGaeq4Xdm galeaacqqHAoWsaeqaniabgUIiYdGccaWGKbGaeqiXdqhaleaacqaH epaDdaWgaaadbaGaeqyVd4gabeaaaSqaaiabes8a0naaBaaameaacq aH9oGBcqGHRaWkcaaIXaaabeaaa0Gaey4kIipakiabgkHiTaqaamaa qafabaWaa8qCaeaadaWdrbqaaiabgEGirlabew8a1naabmaabaGaeq 4XdmMaaiilaiabes8a0bGaayjkaiaawMcaaiabgEna0kabe27aUnaa BaaaleaacqqHAoWscaGGSaGaeq4WdmhabeaakmaabmaabaGaeq4Xdm gacaGLOaGaayzkaaGaeqiTdq2aaSbaaSqaaiabeo7aNbqabaGcdaqa daqaaiabeE8aJbGaayjkaiaawMcaaiaadsgacqaHepaDaSqaaiabeo 8aZbqab0Gaey4kIipaaSqaaiabes8a0naaBaaameaacqaH9oGBaeqa aaWcbaGaeqiXdq3aaSbaaWqaaiabe27aUjabgUcaRiaaigdaaeqaaa qdcqGHRiI8aaWcbaGaeq4WdmNaeyicI4SaeuyLdu0aaSbaaWqaaiab fQ5albqabaaaleqaniabggHiLdGccqGH9aqpaeaadaWdXbqaamaape fabaGaeqOXdO2aaeWaaeaacqaHhpWycaGGSaGaeqiXdqhacaGLOaGa ayzkaaGaamizaiabeE8aJbWcbaGaeuOMdSeabeqdcqGHRiI8aOGaam izaiabes8a0bWcbaGaeqiXdq3aaSbaaWqaaiabe27aUbqabaaaleaa cqaHepaDdaWgaaadbaGaeqyVd4Maey4kaSIaaGymaaqabaaaniabgU IiYdaaaaa@A76F@

2 ( Δ α υ( χ, τ ν ) )= 1 κ 2 τ ν1 τ ν τη τ ( Δ α υ ) ττ ( χ,τ )dσ dτ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOaIy7aaW baaSqabeaacaaIYaaaaOWaaeWaaeaacqqHuoardaahaaWcbeqaaiab eg7aHbaakiabew8a1naabmaabaGaeq4XdmMaaiilaiabes8a0naaBa aaleaacqaH9oGBaeqaaaGccaGLOaGaayzkaaaacaGLOaGaayzkaaGa eyypa0ZaaSaaaeaacaaIXaaabaGaeqOUdS2aaWbaaSqabeaacaaIYa aaaaaakmaapehabaWaa8qCaeaadaqadaqaaiabfs5aenaaCaaaleqa baGaeqySdegaaOGaeqyXduhacaGLOaGaayzkaaWaaSbaaSqaaiabes 8a0jabes8a0bqabaGcdaqadaqaaiabeE8aJjaacYcacqaHepaDaiaa wIcacaGLPaaacaWGKbGaeq4WdmhaleaacqaHepaDcqGHsislcqaH3o aAaeaacqaHepaDa0Gaey4kIipakiaadsgacqaHepaDaSqaaiabes8a 0naaBaaameaacqaH9oGBcqGHsislcaaIXaaabeaaaSqaaiabes8a0n aaBaaameaacqaH9oGBaeqaaaqdcqGHRiI8aaaa@7456@

1 κ Κ 1 ( τ ν τ ν+1 Δυ( χ,τ )dτ )dχ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaeqOUdSgaamaapefabaGaeyOaIy7aaWbaaSqabeaacaaI XaaaaOWaaeWaaeaadaWdXbqaaiabgs5aejabew8a1naabmaabaGaeq 4XdmMaaiilaiabes8a0bGaayjkaiaawMcaaiaadsgacqaHepaDaSqa aiabes8a0naaBaaameaacqaH9oGBaeqaaaWcbaGaeqiXdq3aaSbaaW qaaiabe27aUjabgUcaRiaaigdaaeqaaaqdcqGHRiI8aaGccaGLOaGa ayzkaaGaamizaiabeE8aJbWcbaGaeuOMdSeabeqdcqGHRiI8aaaa@59B2@

1 κ 2 Κ τ ν1 τ ν τ ν1 τ σ σ+κ Δ d 2 υ d λ 2 ( χ,λ )dλ dσ dτ dχ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaeqOUdS2aaWbaaSqabeaacaaIYaaaaaaakmaapefabaWa a8qCaeaadaWdXbqaamaapehabaGaeyiLdq0aaSaaaeaacaWGKbWaaW baaSqabeaacaaIYaaaaOGaeqyXduhabaGaamizaiabeU7aSnaaCaaa leqabaGaaGOmaaaaaaGcdaqadaqaaiabeE8aJjaacYcacqaH7oaBai aawIcacaGLPaaacaWGKbGaeq4UdWgaleaacqaHdpWCaeaacqaHdpWC cqGHRaWkcqaH6oWAa0Gaey4kIipakiaadsgacqaHdpWCaSqaaiabes 8a0naaBaaameaacqaH9oGBcqGHsislcaaIXaaabeaaaSqaaiabes8a 0bqdcqGHRiI8aOGaamizaiabes8a0bWcbaGaeqiXdq3aaSbaaWqaai abe27aUjabgkHiTiaaigdaaeqaaaWcbaGaeqiXdq3aaSbaaWqaaiab e27aUbqabaaaniabgUIiYdGccaWGKbGaeq4XdmgaleaacqqHAoWsae qaniabgUIiYdaaaa@73A5@

υ Γ 1 ( ( 0 Τ | υ ˙ | π( τ ) dτ ) 1 π +1+| υ ¯ | ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaauWaaeaacq aHfpqDaiaawMa7caGLkWoacqGHKjYOcqqHtoWrdaWgaaWcbaGaaGym aaqabaGcdaqadaqaamaabmaabaWaa8qCaeaadaabdaqaaiqbew8a1z aacaaacaGLhWUaayjcSdWaaWbaaSqabeaacqaHapaCdaqadaqaaiab es8a0bGaayjkaiaawMcaaaaakiaadsgacqaHepaDaSqaaiaaicdaae aacqqHKoava0Gaey4kIipaaOGaayjkaiaawMcaamaaCaaaleqabaWa aSaaaeaacaaIXaaabaGaeqiWda3aaWbaaWqabeaacqGHsislaaaaaa aakiabgUcaRiaaigdacqGHRaWkdaabdaqaaiqbew8a1zaaraaacaGL hWUaayjcSdaacaGLOaGaayzkaaaaaa@5EF4@

Ξ ( υ ),ν = 0 Τ ( | υ ˙ ( τ ) | π( τ )2 υ ˙ ( τ ), υ ˙ ( τ ) )dτ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaaWaaeaacu qHEoawgaqbamaabmaabaGaeqyXduhacaGLOaGaayzkaaGaaiilaiab e27aUbGaayzkJiaawQYiaiabg2da9maapehabaWaaeWaaeaadaabda qaaiqbew8a1zaacaWaaeWaaeaacqaHepaDaiaawIcacaGLPaaaaiaa wEa7caGLiWoadaahaaWcbeqaaiabec8aWnaabmaabaGaeqiXdqhaca GLOaGaayzkaaGaeyOeI0IaaGOmaaaakiqbew8a1zaacaWaaeWaaeaa cqaHepaDaiaawIcacaGLPaaacaGGSaGafqyXduNbaiaadaqadaqaai abes8a0bGaayjkaiaawMcaaaGaayjkaiaawMcaaiaadsgacqaHepaD aSqaaiaaicdaaeaacqqHKoava0Gaey4kIipaaaa@630C@

Γ 5 ( 0 Τ | υ ˙ ( τ ) | π( τ ) dτ ) α+1 π MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4KdC0aaS baaSqaaiaaiwdaaeqaaOWaaeWaaeaadaWdXbqaamaaemaabaGafqyX duNbaiaadaqadaqaaiabes8a0bGaayjkaiaawMcaaaGaay5bSlaawI a7amaaCaaaleqabaGaeqiWda3aaeWaaeaacqaHepaDaiaawIcacaGL PaaaaaGccaWGKbGaeqiXdqhaleaacaaIWaaabaGaeuiPdqfaniabgU IiYdaakiaawIcacaGLPaaadaahaaWcbeqaamaalaaabaGaeqySdeMa ey4kaSIaaGymaaqaaiabec8aWnaaCaaameqabaGaeyOeI0caaaaaaa aaaa@5511@

φ ( υ ν ), υ ν υ = 0 Τ [ ( | υ ˙ ν ( τ ) | π( τ )2 υ ˙ ν ( τ ), υ ˙ ν ( τ ) υ ˙ ( τ ) ) +( Φ( τ, υ ν ( τ ) ), υ ν ( τ )υ( τ ) ) ]dτ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaaWaaeaacu aHgpGAgaqbamaabmaabaGaeqyXdu3aaSbaaSqaaiabe27aUbqabaaa kiaawIcacaGLPaaacaGGSaGaeqyXdu3aaSbaaSqaaiabe27aUbqaba GccqGHsislcqaHfpqDaiaawMYicaGLQmcacqGH9aqpdaWdXbqaamaa dmaaeaqabeaadaqadaqaamaaemaabaGafqyXduNbaiaadaWgaaWcba GaeqyVd4gabeaakmaabmaabaGaeqiXdqhacaGLOaGaayzkaaaacaGL hWUaayjcSdWaaWbaaSqabeaacqaHapaCdaqadaqaaiabes8a0bGaay jkaiaawMcaaiabgkHiTiaaikdaaaGccuaHfpqDgaGaamaaBaaaleaa cqaH9oGBaeqaaOWaaeWaaeaacqaHepaDaiaawIcacaGLPaaacaGGSa GafqyXduNbaiaadaWgaaWcbaGaeqyVd4gabeaakmaabmaabaGaeqiX dqhacaGLOaGaayzkaaGaeyOeI0IafqyXduNbaiaadaqadaqaaiabes 8a0bGaayjkaiaawMcaaaGaayjkaiaawMcaaaqaaiabgUcaRmaabmaa baGaey4bIeTaeuOPdy0aaeWaaeaacqaHepaDcaGGSaGaeqyXdu3aaS baaSqaaiabe27aUbqabaGcdaqadaqaaiabes8a0bGaayjkaiaawMca aaGaayjkaiaawMcaaiaacYcacqaHfpqDdaWgaaWcbaGaeqyVd4gabe aakmaabmaabaGaeqiXdqhacaGLOaGaayzkaaGaeyOeI0IaeqyXdu3a aeWaaeaacqaHepaDaiaawIcacaGLPaaaaiaawIcacaGLPaaaaaGaay 5waiaaw2faaiaadsgacqaHepaDaSqaaiaaicdaaeaacqqHKoava0Ga ey4kIipaaaa@95D3@

0 Τ 0 1 ( Γ( συ( τ ) )Γ( 0 ),υ( τ ) )dσ dτ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qCaeaada WdXbqaamaabmaabaGaey4bIeTaeu4KdC0aaeWaaeaacqaHdpWCcqaH fpqDdaqadaqaaiabes8a0bGaayjkaiaawMcaaaGaayjkaiaawMcaai abgkHiTiabgEGirlabfo5ahnaabmaabaGaaGimaaGaayjkaiaawMca aiaacYcacqaHfpqDdaqadaqaaiabes8a0bGaayjkaiaawMcaaaGaay jkaiaawMcaaiaadsgacqaHdpWCaSqaaiaaicdaaeaacaaIXaaaniab gUIiYdGccaWGKbGaeqiXdqhaleaacaaIWaaabaGaeuiPdqfaniabgU IiYdaaaa@5C9D@

( Ω ( | α |+ξ=1 λ ρ 2( β+| α |+ξλ ) | Δ α λ υ τ ξ | 2 + | υ | 2 ) ε 2γτ dχdτ ) 1 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WdrbqaamaabmaabaWaaabCaeaacqaHbpGCdaahaaWcbeqaaiaaikda daqadaqaaiabek7aIjabgUcaRmaaemaabaGaeqySdegacaGLhWUaay jcSdGaey4kaSIaeqOVdGNaeyOeI0Iaeq4UdWgacaGLOaGaayzkaaaa aOWaaqWaaeaacqqHuoardaahaaWcbeqaaiabeg7aHbaakiabeU7aSj abew8a1naaBaaaleaacqaHepaDdaahaaadbeqaaiabe67a4baaaSqa baaakiaawEa7caGLiWoadaahaaWcbeqaaiaaikdaaaGccqGHRaWkda abdaqaaiabew8a1bGaay5bSlaawIa7amaaCaaaleqabaGaaGOmaaaa aeaadaabdaqaaiabeg7aHbGaay5bSlaawIa7aiabgUcaRiabe67a4j abg2da9iaaigdaaeaacqaH7oaBa0GaeyyeIuoaaOGaayjkaiaawMca aiabew7aLnaaCaaaleqabaGaeyOeI0IaaGOmaiabeo7aNjabes8a0b aakiaadsgacqaHhpWycaWGKbGaeqiXdqhaleaacqqHPoWvdaWgaaad baGaeyOhIukabeaaaSqab0Gaey4kIipaaOGaayjkaiaawMcaamaaCa aaleqabaWaaSaaaeaacaaIXaaabaGaaGOmaaaaaaaaaa@7FE2@

| π |,| ρ |=0 μ ( 1 ) | π | Ω α πρ ( ·,τ ) Δ ρ υ Δ Π υ ¯ dχ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaada qadaqaaiabgkHiTiaaigdaaiaawIcacaGLPaaadaahaaWcbeqaamaa emaabaGaeqiWdahacaGLhWUaayjcSdaaaOWaa8quaeaacqaHXoqyda WgaaWcbaGaeqiWdaNaeqyWdihabeaakmaabmaabaGaeS4JPFMaaiil aiabes8a0bGaayjkaiaawMcaaiabfs5aenaaCaaaleqabaGaeqyWdi haaOGaeqyXdu3aa0aaaeaacqqHuoardaahaaWcbeqaaiabfc6aqbaa kiabew8a1baacaWGKbGaeq4XdmgaleaacqqHPoWvaeqaniabgUIiYd aaleaadaabdaqaaiabec8aWbGaay5bSlaawIa7aiaacYcadaabdaqa aiabeg8aYbGaay5bSlaawIa7aiabg2da9iaaicdaaeaacqaH8oqBa0 GaeyyeIuoaaaa@6AB6@

ξ=0 μ1 Γ Β ξ υ ξ υ ¯ ν ξ dσ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaabCaeaada Wdrbqaaiabfk5acnaaBaaaleaacqaH+oaEaeqaaOGaeqyXdu3aaSaa aeaacqGHciITdaahaaWcbeqaaiabe67a4baakiqbew8a1zaaraaaba GaeyOaIyRaeqyVd42aaWbaaSqabeaacqaH+oaEaaaaaOGaamizaiab eo8aZbWcbaGaeu4KdCeabeqdcqGHRiI8aaWcbaGaeqOVdGNaeyypa0 JaaGimaaqaaiabeY7aTjabgkHiTiaaigdaa0GaeyyeIuoaaaa@5501@

Γ 1 Κ [ υ 2 + | gradυ | 2 + | α |=2 | Δ α υ 1 | 2 ]d χ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4KdC0aaS baaSqaaiaaigdaaeqaaOWaa8quaeaadaWadaqaaiabew8a1naaCaaa leqabaGaaGOmaaaakiabgUcaRmaaemaabaGaam4zaiaadkhacaWGHb Gaamizaiabew8a1bGaay5bSlaawIa7amaaCaaaleqabaGaaGOmaaaa kiabgUcaRmaaqafabaWaaqWaaeaacqqHuoardaahaaWcbeqaaiabeg 7aHbaakiabew8a1naaBaaaleaacaaIXaaabeaaaOGaay5bSlaawIa7 amaaCaaaleqabaGaaGOmaaaaaeaadaabdaqaaiabeg7aHbGaay5bSl aawIa7aiabg2da9iaaikdaaeqaniabggHiLdaakiaawUfacaGLDbaa caWGKbGafq4XdmMbauaaaSqaaiqbfQ5alzaafaaabeqdcqGHRiI8aa aa@6167@

0 1 ( χ ψ ζ 10 )dζ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qmaeaada qadaqaaiabeE8aJnaaCaaaleqabaGaeqiYdKhaaOGaeyOeI0YaaSaa aeaacqaH2oGEaeaacaaIXaGaaGimaaaaaiaawIcacaGLPaaacaWGKb GaeqOTdOhaleaacaaIWaaabaGaaGymaaqdcqGHRiI8aaaa@45AF@

0 1 0 1 ψ 2 1dχdψ = 0 π 2 0 1 ρdρdθ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qmaeaada WdXaqaaiaaigdacaWGKbGaeq4XdmMaamizaiabeI8a5bWcbaGaaGim aaqaamaakaaabaGaaGymaiabgkHiTiabeI8a5naaCaaameqabaGaaG OmaaaaaeqaaaqdcqGHRiI8aaWcbaGaaGimaaqaaiaaigdaa0Gaey4k Iipakiabg2da9maapedabaWaa8qmaeaacqaHbpGCcaWGKbGaeqyWdi NaamizaiabeI7aXbWcbaGaaGimaaqaaiaaigdaa0Gaey4kIipaaSqa aiaaicdaaeaadaWcaaqaaiabec8aWbqaaiaaikdaaaaaniabgUIiYd aaaa@581C@

[ cos χ 2× χ ] dχ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaa8qaaeaada WadaqaamaalaaabaGaci4yaiaac+gacaGGZbWaaOaaaeaacqaHhpWy aSqabaaakeaacaaIYaGaey41aq7aaOaaaeaacqaHhpWyaSqabaaaaa GccaGLBbGaayzxaaaaleqabeqdcqGHRiI8aOGaamizaiabeE8aJbaa @45FC@

 

6. Limits

6.1 Simple

lim ν | υ ν υ | π( τ ) =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiabe27aUjabgkziUkabg6HiLcqabaGc daabdaqaaiabew8a1naaBaaaleaacqaH9oGBaeqaaOGaeyOeI0Iaeq yXduhacaGLhWUaayjcSdWaaSbaaSqaaiabec8aWnaabmaabaGaeqiX dqhacaGLOaGaayzkaaaabeaakiabg2da9iaaicdaaaa@4EAC@

lim ν ρ( υ ν υ )=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiabe27aUjabgkziUkabg6HiLcqabaGc cqaHbpGCdaqadaqaaiabew8a1naaBaaaleaacqaH9oGBaeqaaOGaey OeI0IaeqyXduhacaGLOaGaayzkaaGaeyypa0JaaGimaaaa@4992@

lim ν ρ( υ ν )=ρ( υ ν ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiabe27aUjabgkziUkabg6HiLcqabaGc cqaHbpGCdaqadaqaaiabew8a1naaBaaaleaacqaH9oGBaeqaaaGcca GLOaGaayzkaaGaeyypa0JaeqyWdi3aaeWaaeaacqaHfpqDdaWgaaWc baGaeqyVd4gabeaaaOGaayjkaiaawMcaaaaa@4D22@

limsup ν Ξ ( υ ν ) Ξ ( υ ), υ ν υ =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaciGGZbGaaiyDaiaacchaaSqaaiabe27aUjab gkziUkabg6HiLcqabaGcdaaadaqaaiqbf65ayzaafaWaaeWaaeaacq aHfpqDdaWgaaWcbaGaeqyVd4gabeaaaOGaayjkaiaawMcaaiabgkHi Tiqbf65ayzaafaWaaeWaaeaacqaHfpqDaiaawIcacaGLPaaacaGGSa GaeqyXdu3aaSbaaSqaaiabe27aUbqabaGccqGHsislcqaHfpqDaiaa wMYicaGLQmcacqGH9aqpcaaIWaaaaa@5849@

[ Α lim χ+0 φ( χ ) χ ] 1 <λ< [ Β lim χ φ( χ ) χ ] 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacq qHroqqdaWfqaqaaiGacYgacaGGPbGaaiyBaaWcbaGaeq4XdmMaeyOK H4Qaey4kaSIaaGimaaqabaGcdaWcaaqaaiabeA8aQnaabmaabaGaeq 4XdmgacaGLOaGaayzkaaaabaGaeq4XdmgaaaGaay5waiaaw2faamaa CaaaleqabaGaeyOeI0IaaGymaaaakiabgYda8iabeU7aSjabgYda8m aadmaabaGaeuOKdi0aaCbeaeaaciGGSbGaaiyAaiaac2gaaSqaaiab eE8aJjabgkziUkabg6HiLcqabaGcdaWcaaqaaiabeA8aQnaabmaaba Gaeq4XdmgacaGLOaGaayzkaaaabaGaeq4XdmgaaaGaay5waiaaw2fa amaaCaaaleqabaGaeyOeI0IaaGymaaaaaaa@6210@

[ Α lim χ φ( χ ) χ ] 1 <λ< [ Β lim χ+0 φ( χ ) χ ] 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacq qHroqqdaWfqaqaaiGacYgacaGGPbGaaiyBaaWcbaGaeq4XdmMaeyOK H4QaeyOhIukabeaakmaalaaabaGaeqOXdO2aaeWaaeaacqaHhpWyai aawIcacaGLPaaaaeaacqaHhpWyaaaacaGLBbGaayzxaaWaaWbaaSqa beaacqGHsislcaaIXaaaaOGaeyipaWJaeq4UdWMaeyipaWZaamWaae aacqqHsoGqdaWfqaqaaiGacYgacaGGPbGaaiyBaaWcbaGaeq4XdmMa eyOKH4Qaey4kaSIaaGimaaqabaGcdaWcaaqaaiabeA8aQnaabmaaba Gaeq4XdmgacaGLOaGaayzkaaaabaGaeq4XdmgaaaGaay5waiaaw2fa amaaCaaaleqabaGaeyOeI0IaaGymaaaaaaa@6210@

λ lim Δτ0 1 Δτ [ ( Τυ )( τ+Δτ )( Τυ )( τ ) ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdW2aaC beaeaaciGGSbGaaiyAaiaac2gaaSqaaiabgs5aejabes8a0jabgkzi UkaaicdaaeqaaOWaaSaaaeaacaaIXaaabaGaeyiLdqKaeqiXdqhaam aadmaabaWaaeWaaeaacqqHKoavcqaHfpqDaiaawIcacaGLPaaadaqa daqaaiabes8a0jabgUcaRiabgs5aejabes8a0bGaayjkaiaawMcaai abgkHiTmaabmaabaGaeuiPdqLaeqyXduhacaGLOaGaayzkaaWaaeWa aeaacqaHepaDaiaawIcacaGLPaaaaiaawUfacaGLDbaaaaa@5BBD@

lim Δτ0 ( Τ 2 υ )( τ+Δτ )( Τ 2 υ )( τ ) Δτ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiabfs5aejabes8a0jabgkziUkaaicda aeqaaOWaaSaaaeaadaqadaqaaiabfs6aunaaBaaaleaacaaIYaaabe aakiabew8a1bGaayjkaiaawMcaamaabmaabaGaeqiXdqNaey4kaSIa euiLdqKaeqiXdqhacaGLOaGaayzkaaGaeyOeI0YaaeWaaeaacqqHKo avdaWgaaWcbaGaaGOmaaqabaGccqaHfpqDaiaawIcacaGLPaaadaqa daqaaiabes8a0bGaayjkaiaawMcaaaqaaiabfs5aejabes8a0baaaa a@593D@

lim ν 0 1 | ( Τ 1 υ ν )( τ )( Τ 1 υ 0 )( τ ) |dτ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiabe27aUjabgkziUkabg6HiLcqabaGc daWdXbqaamaaemaabaWaaeWaaeaacqqHKoavdaWgaaWcbaGaaGymaa qabaGccqaHfpqDdaWgaaWcbaGaeqyVd4gabeaaaOGaayjkaiaawMca amaabmaabaGaeqiXdqhacaGLOaGaayzkaaGaeyOeI0YaaeWaaeaacq qHKoavdaWgaaWcbaGaaGymaaqabaGccqaHfpqDdaWgaaWcbaGaaGim aaqabaaakiaawIcacaGLPaaadaqadaqaaiabes8a0bGaayjkaiaawM caaaGaay5bSlaawIa7aiaadsgacqaHepaDaSqaaiaaicdaaeaacaaI XaaaniabgUIiYdaaaa@5DC3@

lim χ+0 φ( χ ) χ < ( λΒ ) 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiabeE8aJjabgkziUkabgUcaRiaaicda aeqaaOWaaSaaaeaacqaHgpGAdaqadaqaaiabeE8aJbGaayjkaiaawM caaaqaaiabeE8aJbaacqGH8aapdaqadaqaaiabeU7aSjabfk5acbGa ayjkaiaawMcaamaaCaaaleqabaGaeyOeI0IaaGymaaaaaaa@4C86@

λΑ lim χ φ( χ ) χ >1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4UdWMaeu yKde0aaCbeaeaaciGGSbGaaiyAaiaac2gaaSqaaiabeE8aJjabgkzi Ukabg6HiLcqabaGcdaWcaaqaaiabeA8aQnaabmaabaGaeq4Xdmgaca GLOaGaayzkaaaabaGaeq4Xdmgaaiabg6da+iaaigdaaaa@49BA@

lim χ φ( χ ) χ ( λ Α ¯ ) 1 +ε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiabeE8aJjabgkziUkabg6HiLcqabaGc daWcaaqaaiabeA8aQnaabmaabaGaeq4XdmgacaGLOaGaayzkaaaaba Gaeq4XdmgaaiabgwMiZoaabmaabaGaeq4UdWMafuyKdeKbaebaaiaa wIcacaGLPaaadaahaaWcbeqaaiabgkHiTiaaigdaaaGccqGHRaWkcq aH1oqzaaa@4FC6@

lim χ 0 1 χ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiabeE8aJjabgkziUkaaicdadaahaaad beqaaiabgkHiTaaaaSqabaGcdaWcaaqaaiaaigdaaeaacqaHhpWyaa aaaa@4110@

lim x 0 ( 1 χ + 1 χ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcaaIWaWaaWbaaWqa beaacqGHsislaaaaleqaaOWaaeWaaeaadaWcaaqaaiaaigdaaeaacq aHhpWyaaGaey4kaSYaaSaaaeaacaaIXaaabaGaeq4Xdm2aaWbaaSqa beaacaaIYaaaaaaaaOGaayjkaiaawMcaaaaa@4636@

lim x3 1 χ2 χ3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiaadIhacqGHsgIRcaaIZaaabeaakmaa laaabaGaaGymaiabgkHiTmaakaaabaGaeq4XdmMaeyOeI0IaaGOmaa WcbeaaaOqaaiabeE8aJjabgkHiTiaaiodaaaaaaa@454F@

lim δx0 φ( χ 0 +δχ)φ( χ 0 ) δχ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiabes7aKjaadIhacqGHsgIRcaaIWaaa beaakmaalaaabaGaeqOXdOMaaiikaiabeE8aJnaaBaaaleaacaaIWa aabeaakiabgUcaRiabes7aKjabeE8aJjaacMcacqGHsislcqaHgpGA caGGOaGaeq4Xdm2aaSbaaSqaaiaaicdaaeqaaOGaaiykaaqaaiabes 7aKjabeE8aJbaaaaa@5264@

lim β1 β ( γδ ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiabek7aIjabgkziUkaaigdaaeqaaOWa aSaaaeaacqaHYoGyaeaadaqadaqaaiabeo7aNjabgkHiTiabes7aKb GaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaaaaaa@45AF@

6.2 Complex

limsup ν ( Ξ ( υ ν ) Ξ ( υ ), υ ν υ )0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaciGGZbGaaiyDaiaacchaaSqaaiabe27aUjab gkziUkabg6HiLcqabaGcdaqadaqaaiqbf65ayzaafaWaaeWaaeaacq aHfpqDdaWgaaWcbaGaeqyVd4gabeaaaOGaayjkaiaawMcaaiabgkHi Tiqbf65ayzaafaWaaeWaaeaacqaHfpqDaiaawIcacaGLPaaacaGGSa GaeqyXdu3aaSbaaSqaaiabe27aUbqabaGccqGHsislcqaHfpqDaiaa wIcacaGLPaaacqGHKjYOcaaIWaaaaa@58B1@

limsup ν | υ ¯ ν | ρ + α φ( υ ν )= MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaciGGZbGaaiyDaiaacchaaSqaaiabe27aUjab gkziUkabg6HiLcqabaGcdaabdaqaaiqbew8a1zaaraWaaSbaaSqaai abe27aUbqabaaakiaawEa7caGLiWoadaahaaWcbeqaaiabgkHiTiab eg8aYnaaCaaameqabaGaey4kaScaaSGaeqySdegaaOGaeqOXdO2aae WaaeaacqaHfpqDdaWgaaWcbaGaeqyVd4gabeaaaOGaayjkaiaawMca aiabg2da9iabgkHiTiabg6HiLcaa@57F2@

lim Δτ0 λ Δτ 0 1 [ Ξ( τ+Δτ,σ )Ξ( τ,σ ) ]υ( σ )dσ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiabfs5aejabes8a0jabgkziUkaaicda aeqaaOWaaSaaaeaacqaH7oaBaeaacqqHuoarcqaHepaDaaWaa8qCae aadaWadaqaaiabf65aynaabmaabaGaeqiXdqNaey4kaSIaeuiLdqKa eqiXdqNaaiilaiabeo8aZbGaayjkaiaawMcaaiabgkHiTiabf65ayn aabmaabaGaeqiXdqNaaiilaiabeo8aZbGaayjkaiaawMcaaaGaay5w aiaaw2faaiabew8a1naabmaabaGaeq4WdmhacaGLOaGaayzkaaGaam izaiabeo8aZbWcbaGaaGimaaqaaiaaigdaa0Gaey4kIipaaaa@64DE@

lim ν 0 1 | ( Τ 1 υ ν )( τ ) Τ 1 υ 0 ( τ ) |dτ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiabe27aUjabgkziUkabg6HiLcqabaGc daWdXbqaamaaemaabaWaaeWaaeaacqqHKoavdaWgaaWcbaGaaGymaa qabaGccqaHfpqDdaWgaaWcbaGaeqyVd4gabeaaaOGaayjkaiaawMca amaabmaabaGaeqiXdqhacaGLOaGaayzkaaGaeyOeI0IaeuiPdq1aaS baaSqaaiaaigdaaeqaaOGaeqyXdu3aaSbaaSqaaiaaicdaaeqaaOWa aeWaaeaacqaHepaDaiaawIcacaGLPaaaaiaawEa7caGLiWoacaWGKb GaeqiXdqhaleaacaaIWaaabaGaaGymaaqdcqGHRiI8aaaa@5C3A@

lim ν φ( τ, υ ν ( τ ), υ ν ( τ ), υ ( τ ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiabe27aUjabgkziUkabg6HiLcqabaGc cqaHgpGAdaqadaqaaiabes8a0jaacYcacqaHfpqDdaWgaaWcbaGaeq yVd4gabeaakmaabmaabaGaeqiXdqhacaGLOaGaayzkaaGaaiilaiqb ew8a1zaafaWaaSbaaSqaaiabe27aUbqabaGcdaqadaqaaiabes8a0b GaayjkaiaawMcaaiaacYcacuaHfpqDgaGbamaabmaabaGaeqiXdqha caGLOaGaayzkaaaacaGLOaGaayzkaaaaaa@586F@

lim Δτ0 0 1 Ξ( τ+Δτ )Ξ( τ,σ ) Δτ η( σ ) φ( σ,υ( σ ), υ ( σ ), υ ( σ ) )dσ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiabgs5aejabes8a0jabgkziUkaaicda aeqaaOWaa8qCaqaabeqaamaalaaabaGaeuONdG1aaeWaaeaacqaHep aDcqGHRaWkcqGHuoarcqaHepaDaiaawIcacaGLPaaacqGHsislcqqH Eoawdaqadaqaaiabes8a0jaacYcacqaHdpWCaiaawIcacaGLPaaaae aacqGHuoarcqaHepaDaaGaeq4TdG2aaeWaaeaacqaHdpWCaiaawIca caGLPaaaaeaacqaHgpGAdaqadaqaaiabeo8aZjaacYcacqaHfpqDda qadaqaaiabeo8aZbGaayjkaiaawMcaaiaacYcacuaHfpqDgaqbamaa bmaabaGaeq4WdmhacaGLOaGaayzkaaGaaiilaiqbew8a1zaagaWaae WaaeaacqaHdpWCaiaawIcacaGLPaaaaiaawIcacaGLPaaacaWGKbGa eq4WdmhaaSqaaiaaicdaaeaacaaIXaaaniabgUIiYdaaaa@751F@

Α= lim ν κ=1 ν φ( δ κ )Δχ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuyKdeKaey ypa0ZaaCbeaeaaciGGSbGaaiyAaiaac2gaaSqaaiabe27aUjabgkzi Ukabg6HiLcqabaGcdaaeWbqaaiabeA8aQjaacIcacqaH0oazdaWgaa WcbaGaeqOUdSgabeaakiaacMcacqqHuoarcqaHhpWyaSqaaiabeQ7a Rjabg2da9iaaigdaaeaacqaH9oGBa0GaeyyeIuoaaaa@51B3@

lim χ 0 χ e ψ 2 dψ = π 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaCbeaeaaci GGSbGaaiyAaiaac2gaaSqaaiabeE8aJjabgkziUkabg6HiLcqabaGc daWdXaqaaiaadwgadaahaaWcbeqaaiabgkHiTiabeI8a5naaCaaame qabaGaaGOmaaaaaaGccaWGKbGaeqiYdKhaleaacaaIWaaabaGaeq4X dmganiabgUIiYdGccqGH9aqpdaWcaaqaamaakaaabaGaeqiWdahale qaaaGcbaGaaGOmaaaaaaa@4DF4@

7. Matrices

7.1 Simple

( β 1 β 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafa qabeGabaaabaGaeqOSdi2aaSbaaSqaaiaaigdaaeqaaaGcbaGaeqOS di2aaSbaaSqaaiaaikdaaeqaaaaaaOGaayjkaiaawMcaaaaa@3CB2@

( Σ 1,1 Σ 1,2 Σ 2,1 Σ 2,2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafa qabeGacaaabaGaeu4Odm1aaSbaaSqaaiaaigdacaGGSaGaaGymaaqa baaakeaacqqHJoWudaWgaaWcbaGaaGymaiaacYcacaaIYaaabeaaaO qaaiabfo6atnaaBaaaleaacaaIYaGaaiilaiaaigdaaeqaaaGcbaGa eu4Odm1aaSbaaSqaaiaaikdacaGGSaGaaGOmaaqabaaaaaGccaGLOa Gaayzkaaaaaa@4714@

( Σ 1,1 Σ 1,2 Σ 2,1 Σ 2,2 )= ( Σ 1,1 Σ 1,2 Σ 2,1 Σ 2,2 ) 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafa qabeGacaaabaGaeu4Odm1aaWbaaSqabeaacaaIXaGaaiilaiaaigda aaaakeaacqqHJoWudaahaaWcbeqaaiaaigdacaGGSaGaaGOmaaaaaO qaaiabfo6atnaaCaaaleqabaGaaGOmaiaacYcacaaIXaaaaaGcbaGa eu4Odm1aaWbaaSqabeaacaaIYaGaaiilaiaaikdaaaaaaaGccaGLOa GaayzkaaGaeyypa0ZaaeWaaeaafaqabeGacaaabaGaeu4Odm1aaSba aSqaaiaaigdacaGGSaGaaGymaaqabaaakeaacqqHJoWudaWgaaWcba GaaGymaiaacYcacaaIYaaabeaaaOqaaiabfo6atnaaBaaaleaacaaI YaGaaiilaiaaigdaaeqaaaGcbaGaeu4Odm1aaSbaaSqaaiaaikdaca GGSaGaaGOmaaqabaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacqGH sislcaaIXaaaaaaa@5B10@

( Α Β Β Γ ) 1 =( αα αβ βα ββ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafa qabeGacaaabaGaeuyKdeeabaGaeuOKdieabaGafuOKdiKbauaaaeaa cqqHtoWraaaacaGLOaGaayzkaaWaaWbaaSqabeaacqGHsislcaaIXa aaaOGaeyypa0ZaaeWaaeaafaqabeGacaaabaGaeqySdeMaeqySdega baGaeqySdeMaeqOSdigabaGaeqOSdiMaeqySdegabaGaeqOSdiMaeq OSdigaaaGaayjkaiaawMcaaaaa@4EA6@

( Υ 1 Υ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafa qabeGabaaabaGaeuyPdu1aaSbaaSqaaiaaigdaaeqaaaGcbaGaeuyP du1aaSbaaSqaaiaaikdaaeqaaaaaaOGaayjkaiaawMcaaaaa@3C80@

( σ 1 2 γ γ σ 2 2 ) Ι ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafa qabeGacaaabaGaeq4Wdm3aa0baaSqaaiaaigdaaeaacaaIYaaaaaGc baGaeq4SdCgabaGaeq4SdCgabaGaeq4Wdm3aa0baaSqaaiaaikdaae aacaaIYaaaaaaaaOGaayjkaiaawMcaaiabfM5ajnaaBaaaleaacqaH 9oGBaeqaaaaa@4515@

Β 1 =( 1 0 0 0 ) Ι ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOKdi0aaS baaSqaaiaaigdaaeqaaOGaeyypa0ZaaeWaaeaafaqabeGacaaabaGa aGymaaqaaiaaicdaaeaacaaIWaaabaGaaGimaaaaaiaawIcacaGLPa aacqGHxkcXcqqHzoqsdaWgaaWcbaGaeqyVd4gabeaaaaa@432F@

Β 2 =( 0 0 0 1 ) Ι ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOKdi0aaS baaSqaaiaaikdaaeqaaOGaeyypa0ZaaeWaaeaafaqabeGacaaabaGa aGimaaqaaiaaicdaaeaacaaIWaaabaGaaGymaaaaaiaawIcacaGLPa aacqGHxkcXcqqHzoqsdaWgaaWcbaGaeqyVd4gabeaaaaa@4330@

Β 3 =( 0 1 1 0 ) Ι ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOKdi0aaS baaSqaaiaaiodaaeqaaOGaeyypa0ZaaeWaaeaafaqabeGacaaabaGa aGimaaqaaiaaigdaaeaacaaIXaaabaGaaGimaaaaaiaawIcacaGLPa aacqGHxkcXcqqHzoqsdaWgaaWcbaGaeqyVd4gabeaaaaa@4332@

Μ ( Χ 0 0 Χ ) = Ι 2 Μ Χ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiNd00aaS baaSqaamaabmaabaqbaeqabiGaaaqaaiabfE6adbqaaiaaicdaaeaa caaIWaaabaGaeu4PdmeaaaGaayjkaiaawMcaaaqabaGccqGH9aqpcq qHzoqsdaWgaaWcbaGaaGOmaaqabaGccqGHxkcXcqqHCoqtdaWgaaWc baGaeu4Pdmeabeaaaaa@462D@

var( vec( Υ _ ^ 1 ) vec( Υ _ ^ 2 ) )=( Σ 1,1 Σ 1,2 Σ 2,1 Σ 2,2 ) Ι ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciODaiaacg gacaGGYbWaaeWaaeaafaqabeGabaaabaGaamODaiaadwgacaWGJbWa aeWaaeaacuqHLoqvgaqhgaqcamaaBaaaleaacaaIXaaabeaaaOGaay jkaiaawMcaaaqaaiaadAhacaWGLbGaam4yamaabmaabaGafuyPduLb a0HbaKaadaWgaaWcbaGaaGOmaaqabaaakiaawIcacaGLPaaaaaaaca GLOaGaayzkaaGaeyypa0ZaaeWaaeaafaqabeGacaaabaGaeu4Odm1a aSbaaSqaaiaaigdacaGGSaGaaGymaaqabaaakeaacqqHJoWudaWgaa WcbaGaaGymaiaacYcacaaIYaaabeaaaOqaaiabfo6atnaaBaaaleaa caaIYaGaaiilaiaaigdaaeqaaaGcbaGaeu4Odm1aaSbaaSqaaiaaik dacaGGSaGaaGOmaaqabaaaaaGccaGLOaGaayzkaaGaey4LIqSaeuyM dK0aaSbaaSqaaiabe27aUbqabaaaaa@5FE9@

Ιν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuyMdKKaey 4LIqSaeqyVd4gaaa@3B28@

7.2 Complex

( Υ 1 Υ 2 )[ ( Χ 0 0 Χ )( β 1 β 2 ),( Σ 1,1 Σ 1,2 Σ 2,1 Σ 2,2 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafa qabeGabaaabaGaeuyPdu1aaSbaaSqaaiaaigdaaeqaaaGcbaGaeuyP du1aaSbaaSqaaiaaikdaaeqaaaaaaOGaayjkaiaawMcaaiablYJi6m aadmaabaWaaeWaaeaafaqabeGacaaabaGaeu4PdmeabaGaaGimaaqa aiaaicdaaeaacqqHNoWqaaaacaGLOaGaayzkaaWaaeWaaeaafaqabe GabaaabaGaeqOSdi2aaSbaaSqaaiaaigdaaeqaaaGcbaGaeqOSdi2a aSbaaSqaaiaaikdaaeqaaaaaaOGaayjkaiaawMcaaiaacYcadaqada qaauaabeqaciaaaeaacqqHJoWudaWgaaWcbaGaaGymaiaacYcacaaI XaaabeaaaOqaaiabfo6atnaaBaaaleaacaaIXaGaaiilaiaaikdaae qaaaGcbaGaeu4Odm1aaSbaaSqaaiaaikdacaGGSaGaaGymaaqabaaa keaacqqHJoWudaWgaaWcbaGaaGOmaiaacYcacaaIYaaabeaaaaaaki aawIcacaGLPaaaaiaawUfacaGLDbaaaaa@5E20@

( β ^ 1 β ^ 2 )= [ ( Χ 1 0 0 Χ 2 )( Σ 1,1 Σ 1,2 Σ 2,1 Σ 2,1 )( Χ 1 0 0 Χ 2 ) ] 1 ×( Χ 1 0 0 Χ 2 )( Σ 1,1 Σ 1,2 Σ 2,1 Σ 2,1 )( Υ 1 Υ 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaqada qaauaabeqaceaaaeaacuaHYoGygaqcamaaBaaaleaacaaIXaaabeaa aOqaaiqbek7aIzaajaWaaSbaaSqaaiaaikdaaeqaaaaaaOGaayjkai aawMcaaiabg2da9maadmaabaWaaeWaaeaafaqabeGacaaabaGafu4P dmKbauaadaWgaaWcbaGaaGymaaqabaaakeaacaaIWaaabaGaaGimaa qaaiqbfE6adzaafaWaaSbaaSqaaiaaikdaaeqaaaaaaOGaayjkaiaa wMcaamaabmaabaqbaeqabiGaaaqaaiabfo6atnaaCaaaleqabaGaaG ymaiaacYcacaaIXaaaaaGcbaGaeu4Odm1aaWbaaSqabeaacaaIXaGa aiilaiaaikdaaaaakeaacqqHJoWudaahaaWcbeqaaiaaikdacaGGSa GaaGymaaaaaOqaaiabfo6atnaaCaaaleqabaGaaGOmaiaacYcacaaI XaaaaaaaaOGaayjkaiaawMcaamaabmaabaqbaeqabiGaaaqaaiabfE 6adnaaBaaaleaacaaIXaaabeaaaOqaaiaaicdaaeaacaaIWaaabaGa eu4Pdm0aaSbaaSqaaiaaikdaaeqaaaaaaOGaayjkaiaawMcaaaGaay 5waiaaw2faamaaCaaaleqabaGaeyOeI0IaaGymaaaaaOqaaiabgEna 0oaabmaabaqbaeqabiGaaaqaaiqbfE6adzaafaWaaSbaaSqaaiaaig daaeqaaaGcbaGaaGimaaqaaiaaicdaaeaacuqHNoWqgaqbamaaBaaa leaacaaIYaaabeaaaaaakiaawIcacaGLPaaadaqadaqaauaabeqaci aaaeaacqqHJoWudaahaaWcbeqaaiaaigdacaGGSaGaaGymaaaaaOqa aiabfo6atnaaCaaaleqabaGaaGymaiaacYcacaaIYaaaaaGcbaGaeu 4Odm1aaWbaaSqabeaacaaIYaGaaiilaiaaigdaaaaakeaacqqHJoWu daahaaWcbeqaaiaaikdacaGGSaGaaGymaaaaaaaakiaawIcacaGLPa aadaqadaqaauaabeqaceaaaeaacqqHLoqvdaWgaaWcbaGaaGymaaqa baaakeaacqqHLoqvdaWgaaWcbaGaaGOmaaqabaaaaaGccaGLOaGaay zkaaaaaaa@8460@

[ ( Χ 1 0 Σ 2,1 Σ 1,1 1 Χ 1 Χ 2 )( β 1 β 2 ),( Σ 1,1 0 0 Σ 22.1 ) ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaada qadaqaauaabeqaciaaaeaacqqHNoWqdaWgaaWcbaGaaGymaaqabaaa keaacaaIWaaabaGaeyOeI0Iaeu4Odm1aaSbaaSqaaiaaikdacaGGSa GaaGymaaqabaGccqqHJoWudaqhaaWcbaGaaGymaiaacYcacaaIXaaa baGaeyOeI0IaaGymaaaakiabfE6adnaaBaaaleaacaaIXaaabeaaaO qaaiabfE6adnaaBaaaleaacaaIYaaabeaaaaaakiaawIcacaGLPaaa daqadaqaauaabeqaceaaaeaacqaHYoGydaWgaaWcbaGaaGymaaqaba aakeaacqaHYoGydaWgaaWcbaGaaGOmaaqabaaaaaGccaGLOaGaayzk aaGaaiilamaabmaabaqbaeqabiGaaaqaaiabfo6atnaaBaaaleaaca aIXaGaaiilaiaaigdaaeqaaaGcbaGaaGimaaqaaiaaicdaaeaacqqH JoWudaWgaaWcbaGaaGOmaiaaikdacaGGUaGaaGymaaqabaaaaaGcca GLOaGaayzkaaaacaGLBbGaayzxaaaaaa@5EC6@

( Ι,0 )( Χ 1 Σ 1,1 Χ 1 Χ 1 Σ 1,2 Χ 2 Χ 2 Σ 2,1 Χ 1 Χ 2 Σ 2,2 Χ 2 )( 0 Ι ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq qHzoqscaGGSaGaaGimaaGaayjkaiaawMcaamaabmaabaqbaeqabiGa aaqaaiqbfE6adzaafaWaaSbaaSqaaiaaigdaaeqaaOGaeu4Odm1aaW baaSqabeaacaaIXaGaaiilaiaaigdaaaGccqqHNoWqdaWgaaWcbaGa aGymaaqabaaakeaacuqHNoWqgaqbamaaBaaaleaacaaIXaaabeaaki abfo6atnaaCaaaleqabaGaaGymaiaacYcacaaIYaaaaOGaeu4Pdm0a aSbaaSqaaiaaikdaaeqaaaGcbaGafu4PdmKbauaadaWgaaWcbaGaaG OmaaqabaGccqqHJoWudaahaaWcbeqaaiaaikdacaGGSaGaaGymaaaa kiabfE6adnaaBaaaleaacaaIXaaabeaaaOqaaiqbfE6adzaafaWaaS baaSqaaiaaikdaaeqaaOGaeu4Odm1aaWbaaSqabeaacaaIYaGaaiil aiaaikdaaaGccqqHNoWqdaWgaaWcbaGaaGOmaaqabaaaaaGccaGLOa GaayzkaaWaaeWaaeaafaqabeGabaaabaGaaGimaaqaaiabfM5ajbaa aiaawIcacaGLPaaaaaa@62B7@

( β ˜ 1 β ˜ 2 ) Ν 2 κ [ ( β 1 β 2 ),( Var( β ˜ 1 ) cov( β ˜ 1 , β ˜ 2 ) cov( β ˜ 2 , β ˜ 1 ) Var( β ˜ 2 ) ) ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafa qabeGabaaabaGafqOSdiMbaGaadaWgaaWcbaGaaGymaaqabaaakeaa cuaHYoGygaacamaaBaaaleaacaaIYaaabeaaaaaakiaawIcacaGLPa aacqWI8iIocqqHDoGtdaWgaaWcbaGaaGOmamaaCaaameqabaGaeqOU dSgaaaWcbeaakmaadmaabaWaaeWaaeaafaqabeGabaaabaGaeqOSdi 2aaSbaaSqaaiaaigdaaeqaaaGcbaGaeqOSdi2aaSbaaSqaaiaaikda aeqaaaaaaOGaayjkaiaawMcaaiaacYcadaqadaqaauaabeqaciaaae aacaWGwbGaamyyaiaadkhadaqadaqaaiqbek7aIzaaiaWaaSbaaSqa aiaaigdaaeqaaaGccaGLOaGaayzkaaaabaGaci4yaiaac+gacaGG2b WaaeWaaeaacuaHYoGygaacamaaBaaaleaacaaIXaaabeaakiaacYca cuaHYoGygaacamaaBaaaleaacaaIYaaabeaaaOGaayjkaiaawMcaaa qaaiGacogacaGGVbGaaiODamaabmaabaGafqOSdiMbaGaadaWgaaWc baGaaGOmaaqabaGccaGGSaGafqOSdiMbaGaadaWgaaWcbaGaaGymaa qabaaakiaawIcacaGLPaaaaeaacaWGwbGaamyyaiaadkhadaqadaqa aiqbek7aIzaaiaWaaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaa aaaaGaayjkaiaawMcaaaGaay5waiaaw2faaaaa@6FAE@

( Υ 1 Υ 2 ) Ν 2ν [ ( Ι 2 Χ )( β 1 β 2 ),( σ 1 2 γ γ σ 2 2 ) Ι ν ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafa qabeGabaaabaGaeuyPdu1aaSbaaSqaaiaaigdaaeqaaaGcbaGaeuyP du1aaSbaaSqaaiaaikdaaeqaaaaaaOGaayjkaiaawMcaaiablYJi6i abf25aonaaBaaaleaacaaIYaGaeqyVd4gabeaakmaadmaabaWaaeWa aeaacqqHzoqsdaWgaaWcbaGaaGOmaaqabaGccqGHxkcXcqqHNoWqai aawIcacaGLPaaadaqadaqaauaabeqaceaaaeaacqaHYoGydaWgaaWc baGaaGymaaqabaaakeaacqaHYoGydaWgaaWcbaGaaGOmaaqabaaaaa GccaGLOaGaayzkaaGaaiilamaabmaabaqbaeqabiGaaaqaaiabeo8a ZnaaDaaaleaacaaIXaaabaGaaGOmaaaaaOqaaiabeo7aNbqaaiabeo 7aNbqaaiabeo8aZnaaDaaaleaacaaIYaaabaGaaGOmaaaaaaaakiaa wIcacaGLPaaacqGHxkcXcqqHzoqsdaWgaaWcbaGaeqyVd4gabeaaaO Gaay5waiaaw2faaaaa@63C6@

( σ 2,0 4 γ 0 2 2 γ 0 σ 2,0 2 γ 0 2 σ 1,0 4 2 γ 0 σ 1,0 2 2 γ 0 σ 2,0 2 2 γ 0 σ 1,0 2 2( σ 1,0 2 , σ 2,0 2 + γ 0 2 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafa qabeWadaaabaGaeq4Wdm3aa0baaSqaaiaaikdacaGGSaGaaGimaaqa aiaaisdaaaaakeaacqaHZoWzdaqhaaWcbaGaaGimaaqaaiaaikdaaa aakeaacqGHsislcaaIYaGaeq4SdC2aaSbaaSqaaiaaicdaaeqaaOGa eq4Wdm3aa0baaSqaaiaaikdacaGGSaGaaGimaaqaaiaaikdaaaaake aacqaHZoWzdaqhaaWcbaGaaGimaaqaaiaaikdaaaaakeaacqaHdpWC daqhaaWcbaGaaGymaiaacYcacaaIWaaabaGaaGinaaaaaOqaaiabgk HiTiaaikdacqaHZoWzdaWgaaWcbaGaaGimaaqabaGccqaHdpWCdaqh aaWcbaGaaGymaiaacYcacaaIWaaabaGaaGOmaaaaaOqaaiabgkHiTi aaikdacqaHZoWzdaWgaaWcbaGaaGimaaqabaGccqaHdpWCdaqhaaWc baGaaGOmaiaacYcacaaIWaaabaGaaGOmaaaaaOqaaiabgkHiTiaaik dacqaHZoWzdaWgaaWcbaGaaGimaaqabaGccqaHdpWCdaqhaaWcbaGa aGymaiaacYcacaaIWaaabaGaaGOmaaaaaOqaaiaaikdadaqadaqaai abeo8aZnaaDaaaleaacaaIXaGaaiilaiaaicdaaeaacaaIYaaaaOGa aiilaiabeo8aZnaaDaaaleaacaaIYaGaaiilaiaaicdaaeaacaaIYa aaaOGaey4kaSIaeq4SdC2aa0baaSqaaiaaicdaaeaacaaIYaaaaaGc caGLOaGaayzkaaaaaaGaayjkaiaawMcaaaaa@7D4A@

{ Μ ( Χ 0 0 Χ ) [ ( σ 1,0 2 γ 0 γ 0 σ 2,0 2 ) Ι ν ] Μ ( Χ 0 0 Χ ) } + MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacq qHCoqtdaWgaaWcbaWaaeWaaeaafaqabeGacaaabaGaeu4PdmeabaGa aGimaaqaaiaaicdaaeaacqqHNoWqaaaacaGLOaGaayzkaaaabeaakm aadmaabaWaaeWaaeaafaqabeGacaaabaGaeq4Wdm3aa0baaSqaaiaa igdacaGGSaGaaGimaaqaaiaaikdaaaaakeaacqaHZoWzdaWgaaWcba GaaGimaaqabaaakeaacqaHZoWzdaWgaaWcbaGaaGimaaqabaaakeaa cqaHdpWCdaqhaaWcbaGaaGOmaiaacYcacaaIWaaabaGaaGOmaaaaaa aakiaawIcacaGLPaaacqGHxkcXcqqHzoqsdaWgaaWcbaGaeqyVd4ga beaaaOGaay5waiaaw2faaiabfY5annaaBaaaleaadaqadaqaauaabe qaciaaaeaacqqHNoWqaeaacaaIWaaabaGaaGimaaqaaiabfE6adbaa aiaawIcacaGLPaaaaeqaaaGccaGL7bGaayzFaaWaaWbaaSqabeaacq GHRaWkaaaaaa@6062@

( η ( Μ Α Σ 0 Μ Α ) + Β 1 ( Μ Α Σ 0 Μ Α ) + η η ( Μ Α Σ 0 Μ Α ) + Β π ( Μ Α Σ 0 Μ Α ) + η ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaqaabe qaaiqbeE7aOzaafaWaaeWaaeaacqqHCoqtdaWgaaWcbaGaeuyKdeea beaakiabfo6atnaaBaaaleaacaaIWaaabeaakiabfY5annaaBaaale aacqqHroqqaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacqGHRaWk aaGccqqHsoGqdaWgaaWcbaGaaGymaaqabaGcdaqadaqaaiabfY5ann aaBaaaleaacqqHroqqaeqaaOGaeu4Odm1aaSbaaSqaaiaaicdaaeqa aOGaeuiNd00aaSbaaSqaaiabfg5abbqabaaakiaawIcacaGLPaaada ahaaWcbeqaaiabgUcaRaaakiabeE7aObqaaiabl6UinbqaaiqbeE7a OzaafaWaaeWaaeaacqqHCoqtdaWgaaWcbaGaeuyKdeeabeaakiabfo 6atnaaBaaaleaacaaIWaaabeaakiabfY5annaaBaaaleaacqqHroqq aeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacqGHRaWkaaGccqqHso GqdaWgaaWcbaGaeqiWdahabeaakmaabmaabaGaeuiNd00aaSbaaSqa aiabfg5abbqabaGccqqHJoWudaWgaaWcbaGaaGimaaqabaGccqqHCo qtdaWgaaWcbaGaeuyKdeeabeaaaOGaayjkaiaawMcaamaaCaaaleqa baGaey4kaScaaOGaeq4TdGgaaiaawIcacaGLPaaaaaa@72A6@

Κ=( φ 1 ,, φ ρ( Τ ) )( 1 λ 1 0 0 1 λ ρ( Τ ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOMdSKaey ypa0ZaaeWaaeaacqaHgpGAdaWgaaWcbaGaaGymaaqabaGccaGGSaGa eSOjGSKaaiilaiabeA8aQnaaBaaaleaacqaHbpGCdaqadaqaaiabfs 6aubGaayjkaiaawMcaaaqabaaakiaawIcacaGLPaaadaqadaqaauaa beqadmaaaeaadaWcaaqaaiaaigdaaeaadaGcaaqaaiabeU7aSnaaBa aaleaacaaIXaaabeaaaeqaaaaaaOqaaiablAcilbqaaiaaicdaaeaa cqWIUlstaeaacqWIXlYtaeaacqWIUlstaeaacaaIWaaabaGaeS47IW eabaWaaSaaaeaacaaIXaaabaWaaOaaaeaacqaH7oaBdaWgaaWcbaGa eqyWdi3aaeWaaeaacqqHKoavaiaawIcacaGLPaaaaeqaaaqabaaaaa aaaOGaayjkaiaawMcaaaaa@5CF2@

( Σ γ τ(γ) )=( 1 1 1 1 ζ 2 ζ 1 ζ ζ 2 )×( θ σ(θ) σ 2 (θ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaafa qabeWabaaabaGaeu4OdmfabaGaeq4SdCgabaGaeqiXdqNaaiikaiab eo7aNjaacMcaaaaacaGLOaGaayzkaaGaeyypa0ZaaeWaaeaafaqabe WadaaabaGaaGymaaqaaiaaigdaaeaacaaIXaaabaGaaGymaaqaaiab eA7a6naaCaaaleqabaGaaGOmaaaaaOqaaiabeA7a6bqaaiaaigdaae aacqaH2oGEaeaacqaH2oGEdaahaaWcbeqaaiaaikdaaaaaaaGccaGL OaGaayzkaaGaey41aq7aaeWaaeaafaqabeWabaaabaGaeqiUdehaba Gaeq4WdmNaaiikaiabeI7aXjaacMcaaeaacqaHdpWCdaahaaWcbeqa aiaaikdaaaGccaGGOaGaeqiUdeNaaiykaaaaaiaawIcacaGLPaaaaa a@5EA2@

8. Sets

8.1 Simple

Τ( 0, ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiPdqLaey icI48aaeWaaeaacaaIWaGaaiilaiabg6HiLcGaayjkaiaawMcaaaaa @3D65@

ω=( ω 1 ,, ω ν1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyYdCNaey ypa0ZaaeWaaeaacqaHjpWDdaWgaaWcbaGaaGymaaqabaGccaGGSaGa eSOjGSKaaiilaiabeM8a3naaBaaaleaacqaH9oGBcqGHsislcaaIXa aabeaaaOGaayjkaiaawMcaaaaa@44F6@

Α =Α×( 0, ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuyKde0aaS baaSqaaiabg6HiLcqabaGccqGH9aqpcqqHroqqcqGHxdaTdaqadaqa aiaaicdacaGGSaGaeyOhIukacaGLOaGaayzkaaaaaa@41DF@

Β χ ( 0,; ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOKdi0aaW baaSqabeaacqaHhpWyaaGcdaqadaqaaiaaicdacaGGSaGaeyOhIuQa ai4oaiablkqiJcGaayjkaiaawMcaaaaa@3FC1@

δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHe6aaW baaSqabeaacqaH0oazaaaaaa@3939@

δ\{ 0 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdqMaey icI4SaeSyfHuQaaiixamaacmaabaGaaGimaaGaay5Eaiaaw2haaaaa @3E57@

σΩ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaey OGIWSaeyOaIyRaeyyQdCfaaa@3CAB@

[ χ κ ,χ ]Κ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaamWaaeaacq aHhpWydaWgaaWcbaGaeqOUdSgabeaakiaacYcacqaHhpWyaiaawUfa caGLDbaacqGHckcZcqqHAoWsaaa@415E@

χ Ω ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4XdmMaey icI4SafuyQdCLbaebaaaa@3AD8@

Μ σ ={ Κ,Λ } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiNd00aaS baaSqaaiabeo8aZbqabaGccqGH9aqpdaGadaqaaiabfQ5aljaacYca cqqHBoataiaawUhacaGL9baaaaa@4036@

Γ ,α ( Ω×[ 0, ] ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4KdC0aaW baaSqabeaacqGHEisPcaGGSaGaeqySdegaaOWaaeWaaeaacqGHciIT cqqHPoWvcqGHxdaTdaWadaqaaiaaicdacaGGSaGaeyOhIukacaGLBb GaayzxaaaacaGLOaGaayzkaaaaaa@46B7@

χΩ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4XdmMaey icI4SaeyOaIyRaeuyQdCfaaa@3C26@

{ β σ Κ :σΒ,ΚΜ } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacq aHYoGydaqhaaWcbaGaeq4WdmhabaGaeuOMdSeaaOGaaiOoaiabeo8a ZjabgIGiolabfk5acjaacYcacqqHAoWscqGHiiIZcqqHCoqtaiaawU hacaGL9baaaaa@47BB@

{ Κ Κ ν :ν 0,Ν+1,ΚΜ } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaacq qHAoWsdaqhaaWcbaGaeuOMdSeabaGaeqyVd4gaaOGaaiOoaiabe27a UjabgIGiopaaimaabaGaaGimaiaacYcacqqHDoGtcqGHRaWkcaaIXa aacaGLAaJaay5gWaGaaiilaiabfQ5aljabgIGiolabfY5anbGaay5E aiaaw2haaaaa@4C69@

τ 0 [ 0, ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aaS baaSqaaiaaicdaaeqaaOGaeyicI48aamWaaeaacaaIWaGaaiilaiab g6HiLcGaay5waiaaw2faaaaa@3EFD@

υ 0 Λ 2 ( Ω ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyXdu3aaS baaSqaaiaaicdaaeqaaOGaeyicI4Saeu4MdW0aaWbaaSqabeaacaaI YaaaaOWaaeWaaeaacqGHPoWvaiaawIcacaGLPaaaaaa@3FB2@

φ 0 Γ ˙ ( Υ 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdO2aaS baaSqaaiaaicdaaeqaaOGaeyicI4Safu4KdCKbaiaadaahaaWcbeqa aiabg6HiLcaakmaabmaabaGaeuyPdu1aaSbaaSqaaiaaicdaaeqaaa GccaGLOaGaayzkaaaaaa@4142@

Η 2 ( ε γ×τ , Ω ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4LdG0aaW baaSqabeaacaaIYaaaaOWaaeWaaeaacqaH1oqzdaahaaWcbeqaaiab gkHiTiabeo7aNjabgEna0kabes8a0baakiaacYcacqqHPoWvdaWgaa WcbaGaeyOhIukabeaaaOGaayjkaiaawMcaaaaa@4613@

Ω ρ :={ χΩ: ρ 2 <| χ |<2ρ,ρ>0 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuyQdC1aaW baaSqabeaacqaHbpGCaaGccaGG6aGaeyypa0ZaaiWaaeaacqaHhpWy cqGHiiIZcqqHPoWvcaGG6aWaaSaaaeaacqaHbpGCaeaacaaIYaaaai abgYda8maaemaabaGaeq4XdmgacaGLhWUaayjcSdGaeyipaWJaaGOm aiabeg8aYjaacYcacqaHbpGCcqGH+aGpcaaIWaaacaGL7bGaayzFaa aaaa@5313@

Κ={ χ=( χ 1 , χ 2 ) 2 :ρ>0,0<ω< ω 0 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOMdSKaey ypa0ZaaiWaaeaacqaHhpWycqGH9aqpdaqadaqaaiabeE8aJnaaBaaa leaacaaIXaaabeaakiaacYcacqaHhpWydaWgaaWcbaGaaGOmaaqaba aakiaawIcacaGLPaaacqGHiiIZcqWIDesOdaahaaWcbeqaaiaaikda aaGccaGG6aGaeqyWdiNaeyOpa4JaaGimaiaacYcacaaIWaGaeyipaW JaeqyYdCNaeyipaWJaeqyYdC3aaSbaaSqaaiaaicdaaeqaaaGccaGL 7bGaayzFaaaaaa@550B@

Α={ τ[ 0,Τ ]:| υ ˙ ( τ ) |1 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuyKdeKaey ypa0ZaaiWaaeaacqaHepaDcqGHiiIZdaWadaqaaiaaicdacaGGSaGa euiPdqfacaGLBbGaayzxaaGaaiOoamaaemaabaGafqyXduNbaiaada qadaqaaiabes8a0bGaayjkaiaawMcaaaGaay5bSlaawIa7aiabgwMi ZkaaigdaaiaawUhacaGL9baaaaa@4E38@

η:( 0,1 )[ 0, ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4TdGMaai OoamaabmaabaGaaGimaiaacYcacaaIXaaacaGLOaGaayzkaaGaeyOK H46aaKGeaeaacaaIWaGaaiilaiabg6HiLcGaay5waiaawMcaaaaa@42AA@

φ:( 0,1 )×( 0, )×( 0, )×( ,0 )[ 0, ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqOXdOMaai OoamaabmaabaGaaGimaiaacYcacaaIXaaacaGLOaGaayzkaaGaey41 aq7aaeWaaeaacaaIWaGaaiilaiabg6HiLcGaayjkaiaawMcaaiabgE na0oaabmaabaGaaGimaiaacYcacqGHEisPaiaawIcacaGLPaaacqGH xdaTdaqadaqaaiabgkHiTiabg6HiLkaacYcacaaIWaaacaGLOaGaay zkaaGaeyOKH46aaKGeaeaacaaIWaGaaiilaiabg6HiLcGaay5waiaa wMcaaaaa@5719@

Γ 2 [ 0,1 ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4KdC0aaW baaSqabeaacaaIYaaaaOWaamWaaeaacaaIWaGaaiilaiaaigdaaiaa wUfacaGLDbaaaaa@3C69@

Κ( ρ )={ υΚ:| υ |<ρ } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOMdS0aae WaaeaacqaHbpGCaiaawIcacaGLPaaacqGH9aqpdaGadaqaaiabew8a 1jabgIGiolabfQ5aljaacQdadaabdaqaamaafmaabaGaeqyXduhaca GLjWUaayPcSdaacaGLhWUaayjcSdGaeyipaWJaeqyWdihacaGL7bGa ayzFaaaaaa@4E3A@

χ ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4XdmMaey icI4SaeSyhHe6aaWbaaSqabeaacqaH9oGBaaaaaa@3C87@

α[ 0, π 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey icI48aaKGeaeaacaaIWaGaaiilaiabec8aWnaaCaaaleqabaGaeyOe I0caaOGaeyOeI0IaaGymaaGaay5waiaawMcaaaaa@40E0@

υ Κ( ρ 2 ) ¯ \Κ( ρ 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyXduNaey icI48aa0aaaeaacqqHAoWsdaqadaqaaiabeg8aYnaaBaaaleaacaaI YaaabeaaaOGaayjkaiaawMcaaaaacaGGCbGaeuOMdS0aaeWaaeaacq aHbpGCdaWgaaWcbaGaaGymaaqabaaakiaawIcacaGLPaaaaaa@458D@

Κ β ( ρ )={ υ Κ β : υ <ρ } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOMdS0aaW baaSqabeaacqaHYoGyaaGcdaqadaqaaiabeg8aYbGaayjkaiaawMca aiabg2da9maacmaabaGaeqyXduNaeyicI4SaeuOMdS0aaWbaaSqabe aacqaHYoGyaaGccaGG6aWaauWaaeaacqaHfpqDaiaawMa7caGLkWoa cqGH8aapcqaHbpGCaiaawUhacaGL9baaaaa@4EC8@

0 Ω 1 Ω ¯ 1 Ω 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGimaiabgI GiolabfM6axnaaBaaaleaacaaIXaaabeaakiabgkOimlqbfM6axzaa raWaaSbaaSqaaiaaigdaaeqaaOGaeyOGIWSaeuyQdC1aaSbaaSqaai aaikdaaeqaaaaa@43B9@

υ * Κ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyXdu3aaW baaSqabeaacaGGQaaaaOGaeyicI4SaeuOMdSeaaa@3B9A@

Γ( 0,σ )0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4KdC0aae WaaeaacaaIWaGaaiilaiabeo8aZbGaayjkaiaawMcaaiabggMi6kaa icdaaaa@3E98@

Υ 1 ( Χ β1 , Σ 1,1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuyPdu1aaS baaSqaaiaaigdaaeqaaOGaeSipIOZaaeWaaeaacqqHNoWqdaWgaaWc baGaeqOSdiMaaGymaaqabaGccaGGSaGaeu4Odm1aaSbaaSqaaiaaig dacaGGSaGaaGymaaqabaaakiaawIcacaGLPaaaaaa@43BC@

( Υ 1 , Υ 2 )= Σ 1,2 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq qHLoqvdaWgaaWcbaGaaGymaaqabaGccaGGSaGaeuyPdu1aaSbaaSqa aiaaikdaaeqaaaGccaGLOaGaayzkaaGaeyypa0Jaeu4Odm1aaSbaaS qaaiaaigdacaGGSaGaaGOmaaqabaGccqGHGjsUcaaIWaaaaa@448B@

Ω ¯ = ΚΜ Κ ¯ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafuyQdCLbae bacqGH9aqpdaWeqbqaaiqbfQ5alzaaraaaleaacqqHAoWscqGHiiIZ cqqHCoqtaeqaniablQIivbaaaa@4050@

max{ 1,2,3 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyBaiaacg gacaGG4bWaaiWaaeaacaaIXaGaaiilaiaaikdacaGGSaGaaG4maaGa ay5Eaiaaw2haaaaa@3E90@

sup{ 1,2,3 }=3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4Caiaacw hacaGGWbWaaiWaaeaacaaIXaGaaiilaiaaikdacaGGSaGaaG4maaGa ay5Eaiaaw2haaiabg2da9iaaiodaaaa@4065@

ΑΒ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuyKdeKaey OGIWSaeuOKdieaaa@3AB5@

inf{ 1,2,3 }=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyAaiaac6 gacaGGMbWaaiWaaeaacaaIXaGaaiilaiaaikdacaGGSaGaaG4maaGa ay5Eaiaaw2haaiabg2da9iaaigdaaaa@4048@

α Α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaambuaeaacq qHroqqaSqaaiabeg7aHbqab0GaeSOkIufaaaa@3AAA@

α Α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaebuaeaacq qHroqqaSqaaiabeg7aHbqab0Gaey4dIunaaaa@3B08@

αβ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaeS OkIuLaeqOSdigaaa@3A69@

Α= MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuyKdeKaey ypa0JaeyybIymaaa@39D6@

αβ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey OkIGSaeqOSdigaaa@3AD7@

cov( β 1 , β 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGG2bWaaeWaaeaacqaHYoGydaWgaaWcbaGaaGymaaqabaGccaGG SaGaeqOSdi2aaSbaaSqaaiaaikdaaeqaaaGccaGLOaGaayzkaaaaaa@402B@

Var( β 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOvaiaadg gacaWGYbWaaeWaaeaacqaHYoGydaWgaaWcbaGaaGymaaqabaaakiaa wIcacaGLPaaaaaa@3CCA@

χΧ:ψΨ:χ=ψ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyiaIiIaeq 4XdmMaeyicI4Saeu4PdmKaaiOoaiabgoGiKiabeI8a5jabgIGiolab fI6azjaacQdacqaHhpWycqGH9aqpcqaHipqEaaa@4737@

 

8.2 Complex

Γ 0 2 [ 0,1 ]={ υ Γ 2 [ 0,1 ]:υ( 0 )= υ ( 1 )=0 } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeu4KdC0aa0 baaSqaaiaaicdaaeaacaaIYaaaaOWaamWaaeaacaaIWaGaaiilaiaa igdaaiaawUfacaGLDbaacqGH9aqpdaGadaqaaiabew8a1jabgIGiol abfo5ahnaaCaaaleqabaGaaGOmaaaakmaadmaabaGaaGimaiaacYca caaIXaaacaGLBbGaayzxaaGaaiOoaiabew8a1naabmaabaGaaGimaa GaayjkaiaawMcaaiabg2da9iqbew8a1zaafaWaaeWaaeaacaaIXaaa caGLOaGaayzkaaGaeyypa0JaaGimaaGaay5Eaiaaw2haaaaa@55BC@

max{ 1 5 χ 2 + 1 χ : 1 2 ρτ( 3 τ 2 )χρτ( 2τ ) } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyBaiaacg gacaGG4bWaaiWaaeaadaWcaaqaaiaaigdaaeaacaaI1aaaaiabeE8a JnaaCaaaleqabaGaaGOmaaaakiabgUcaRmaalaaabaGaaGymaaqaam aakaaabaGaeq4XdmgaleqaaaaakiaacQdadaWcaaqaaiaaigdaaeaa caaIYaaaaiabeg8aYjabes8a0naabmaabaGaaG4maiabgkHiTiabes 8a0naaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaaiabgsMiJkab eE8aJjabgsMiJkabeg8aYjabes8a0naabmaabaGaaGOmaiabgkHiTi abes8a0bGaayjkaiaawMcaaaGaay5Eaiaaw2haaaaa@5C0B@

min{ 1 5 χ 2 + 1 χ : 1 2 ρτ( 3 τ 2 )χρτ( 2τ ) } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyBaiaacM gacaGGUbWaaiWaaeaadaWcaaqaaiaaigdaaeaacaaI1aaaaiabeE8a JnaaCaaaleqabaGaaGOmaaaakiabgUcaRmaalaaabaGaaGymaaqaam aakaaabaGaeq4XdmgaleqaaaaakiaacQdadaWcaaqaaiaaigdaaeaa caaIYaaaaiabeg8aYjabes8a0naabmaabaGaaG4maiabgkHiTiabes 8a0naaCaaaleqabaGaaGOmaaaaaOGaayjkaiaawMcaaiabgsMiJkab eE8aJjabgsMiJkabeg8aYjabes8a0naabmaabaGaaGOmaiabgkHiTi abes8a0bGaayjkaiaawMcaaaGaay5Eaiaaw2haaaaa@5C09@

υ τ ξ ={ ξ υ 1 τ ξ ,, ξ υ σ τ ξ } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyXdu3aaS baaSqaaiabes8a0naaCaaameqabaGaeqOVdGhaaaWcbeaakiabg2da 9maacmaabaWaaSaaaeaacqGHciITdaahaaWcbeqaaiabe67a4baaki abew8a1naaBaaaleaacaaIXaaabeaaaOqaaiabgkGi2kabes8a0naa CaaaleqabaGaeqOVdGhaaaaakiaacYcacqWIMaYscaGGSaWaaSaaae aacqGHciITdaahaaWcbeqaaiabe67a4baakiabew8a1naaBaaaleaa cqaHdpWCaeqaaaGcbaGaeyOaIyRaeqiXdq3aaWbaaSqabeaacqaH+o aEaaaaaaGccaGL7bGaayzFaaaaaa@5910@

inf{ β Η β λ ( Ω ) :β Η β λ ( Ω ),β|Γ=υ } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyAaiaac6 gacaGGMbWaaiWaaeaadaqbdaqaaiabek7aIbGaayzcSlaawQa7amaa BaaaleaacqqHxoasdaqhaaadbaGaeqOSdigabaGaeq4UdWgaaSWaae WaaeaacqqHPoWvaiaawIcacaGLPaaaaeqaaOGaaiOoaiabek7aIjab gIGiolabfE5ainaaDaaaleaacqaHYoGyaeaacqaH7oaBaaGcdaqada qaaiabfM6axbGaayjkaiaawMcaaiaacYcacqaHYoGydaabbaqaaiab fo5ahbGaay5bSdGaeyypa0JaeqyXduhacaGL7bGaayzFaaaaaa@5C10@

Β δ 0 1,η λ 2 2μ,0 ( ε γτ , Κ ) Β δ 1 ,η λ 2 2μ,0 ( ε γτ , Κ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuOKdi0aa0 baaSqaaiabes7aKnaaBaaameaacaaIWaaabeaaliabgkHiTiaaigda caGGSaGaeq4TdGgabaGaeq4UdW2aaSbaaWqaaiaaikdaaeqaaSGaey OeI0IaaGOmaiabeY7aTjaacYcacaaIWaaaaOWaaeWaaeaacqaH1oqz daahaaWcbeqaaiabgkHiTiabeo7aNjabes8a0baakiaacYcacqqHAo WsdaWgaaWcbaGaeyOhIukabeaaaOGaayjkaiaawMcaaiabgkOimlab fk5acnaaDaaaleaacqaH0oazdaWgaaadbaGaaGymaaqabaWccaGGSa Gaeq4TdGgabaGaeq4UdW2aaSbaaWqaaiaaikdaaeqaaSGaeyOeI0Ia aGOmaiabeY7aTjaacYcacaaIWaaaaOWaaeWaaeaacqaH1oqzdaahaa WcbeqaaiabgkHiTiabeo7aNjabes8a0baakiaacYcacqqHAoWsdaWg aaWcbaGaeyOhIukabeaaaOGaayjkaiaawMcaaaaa@6CAA@

υ( ·,τ ) Β μ 2μ ( Κ ) Β β 2μ ( Κ ) Β β+ε 2μ ( Κ ) Η β+ε 2μ ( Κ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyXdu3aae WaaeaacqWIpM+zcaGGSaGaeqiXdqhacaGLOaGaayzkaaGaeyicI4Sa euOKdi0aa0baaSqaaiabeY7aTbqaaiaaikdacqaH8oqBaaGcdaqada qaaiabfQ5albGaayjkaiaawMcaaiabgkOimlabfk5acnaaDaaaleaa cqaHYoGyaeaacaaIYaGaeqiVd0gaaOWaaeWaaeaacqqHAoWsaiaawI cacaGLPaaacqGHckcZcqqHsoGqdaqhaaWcbaGaeqOSdiMaey4kaSIa eqyTdugabaGaaGOmaiabeY7aTbaakmaabmaabaGaeuOMdSeacaGLOa GaayzkaaGaeyyyIORaeu4LdG0aa0baaSqaaiabek7aIjabgUcaRiab ew7aLbqaaiaaikdacqaH8oqBaaGcdaqadaqaaiabfQ5albGaayjkai aawMcaaaaa@6D43@

9. Trigonometry

9.1 Simple

tanΑ= sinΑ cosΑ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiDaiaacg gacaGGUbGaeuyKdeKaeyypa0ZaaSaaaeaaciGGZbGaaiyAaiaac6ga cqqHroqqaeaaciGGJbGaai4BaiaacohacqqHroqqaaaaaa@43A9@

cotΑ= 1 tanΑ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGG0bGaeuyKdeKaeyypa0ZaaSaaaeaacaaIXaaabaGaciiDaiaa cggacaGGUbGaeuyKdeeaaaaa@402D@

secΑ= 1 cosΑ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4Caiaacw gacaGGJbGaeuyKdeKaeyypa0ZaaSaaaeaacaaIXaaabaGaci4yaiaa c+gacaGGZbGaeuyKdeeaaaaa@4024@

cscΑ= 1 sinΑ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaaco hacaGGJbGaeuyKdeKaeyypa0ZaaSaaaeaacaaIXaaabaGaci4Caiaa cMgacaGGUbGaeuyKdeeaaaaa@4027@

sec 2 Α tan 2 Α=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4Caiaacw gacaGGJbWaaWbaaSqabeaacaaIYaaaaOGaeuyKdeKaeyOeI0IaciiD aiaacggacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaeuyKdeKaeyypa0 JaaGymaaaa@42E5@

tan(Α)=tanΑ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiDaiaacg gacaGGUbGaaiikaiabgkHiTiabfg5abjaacMcacqGH9aqpcqGHsisl ciGG0bGaaiyyaiaac6gacqqHroqqaaa@4292@

cos(Α)=cosΑ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbGaaiikaiabgkHiTiabfg5abjaacMcacqGH9aqpciGGJbGa ai4BaiaacohacqqHroqqaaa@41A9@

sin2Α=2sinΑcosΑ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacM gacaGGUbGaaGOmaiabfg5abjabg2da9iaaikdaciGGZbGaaiyAaiaa c6gacqqHroqqciGGJbGaai4BaiaacohacqqHroqqaaa@4518@

cos2Α= cos 2 Α sin 2 Α =12 sin 2 Α=2 cos 2 Α1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaaciGGJb Gaai4BaiaacohacaaIYaGaeuyKdeKaeyypa0Jaci4yaiaac+gacaGG ZbWaaWbaaSqabeaacaaIYaaaaOGaeuyKdeKaeyOeI0Iaci4CaiaacM gacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaeuyKdeeabaGaeyypa0Ja aGymaiabgkHiTiaaikdaciGGZbGaaiyAaiaac6gadaahaaWcbeqaai aaikdaaaGccqqHroqqcqGH9aqpcaaIYaGaci4yaiaac+gacaGGZbWa aWbaaSqabeaacaaIYaaaaOGaeuyKdeKaeyOeI0IaaGymaaaaaa@5856@

sin3Α=3sinΑ4 sin 3 Α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacM gacaGGUbGaaG4maiabfg5abjabg2da9iaaiodaciGGZbGaaiyAaiaa c6gacqqHroqqcqGHsislcaaI0aGaci4CaiaacMgacaGGUbWaaWbaaS qabeaacaaIZaaaaOGaeuyKdeeaaa@47BE@

sin 2 Α= 1 2 1 2 cos2Α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacM gacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaeuyKdeKaeyypa0ZaaSaa aeaacaaIXaaabaGaaGOmaaaacqGHsisldaWcaaqaaiaaigdaaeaaca aIYaaaaiGacogacaGGVbGaai4CaiaaikdacqqHroqqaaa@4512@

sin 4 Α= 3 8 1 2 cos2Α+ 1 8 cos4Α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacM gacaGGUbWaaWbaaSqabeaacaaI0aaaaOGaeuyKdeKaeyypa0ZaaSaa aeaacaaIZaaabaGaaGioaaaacqGHsisldaWcaaqaaiaaigdaaeaaca aIYaaaaiGacogacaGGVbGaai4CaiaaikdacqqHroqqcqGHRaWkdaWc aaqaaiaaigdaaeaacaaI4aaaaiGacogacaGGVbGaai4Caiaaisdacq qHroqqaaa@4C7C@

cos 5 Α= 5 8 cosΑ+ 5 16 cos3Α+ 1 16 cos5Α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbWaaWbaaSqabeaacaaI1aaaaOGaeuyKdeKaeyypa0ZaaSaa aeaacaaI1aaabaGaaGioaaaaciGGJbGaai4BaiaacohacqqHroqqcq GHRaWkdaWcaaqaaiaaiwdaaeaacaaIXaGaaGOnaaaaciGGJbGaai4B aiaacohacaaIZaGaeuyKdeKaey4kaSYaaSaaaeaacaaIXaaabaGaaG ymaiaaiAdaaaGaci4yaiaac+gacaGGZbGaaGynaiabfg5abbaa@5220@

sinΑ+sinΒ=2sin 1 2 ( Α+Β )cos 1 2 ( ΑΒ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacM gacaGGUbGaeuyKdeKaey4kaSIaci4CaiaacMgacaGGUbGaeuOKdiKa eyypa0JaaGOmaiGacohacaGGPbGaaiOBamaalaaabaGaaGymaaqaai aaikdaaaWaaeWaaeaacqqHroqqcqGHRaWkcqqHsoGqaiaawIcacaGL PaaaciGGJbGaai4BaiaacohadaWcaaqaaiaaigdaaeaacaaIYaaaam aabmaabaGaeuyKdeKaeyOeI0IaeuOKdieacaGLOaGaayzkaaaaaa@542B@

cos 1 ( χ )=π cos 1 χ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbWaaWbaaSqabeaacqGHsislcaaIXaaaaOWaaeWaaeaacqGH sislcqaHhpWyaiaawIcacaGLPaaacqGH9aqpcqaHapaCcqGHsislci GGJbGaai4BaiaacohadaahaaWcbeqaaiabgkHiTiaaigdaaaGccqaH hpWyaaa@48EF@

cos( χ×ψ )+ tan 2 χ+ψ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbWaaeWaaeaacqaHhpWycqGHxdaTcqaHipqEaiaawIcacaGL PaaacqGHRaWkciGG0bGaaiyyaiaac6gadaahaaWcbeqaaiaaikdaaa GccqaHhpWycqGHRaWkcqaHipqEaaa@48FC@

cos( χ×ψ )+ tan 2 ( χ+ψ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbWaaeWaaeaacqaHhpWycqGHxdaTcqaHipqEaiaawIcacaGL PaaacqGHRaWkciGG0bGaaiyyaiaac6gadaahaaWcbeqaaiaaikdaaa GcdaqadaqaaiabeE8aJjabgUcaRiabeI8a5bGaayjkaiaawMcaaaaa @4A85@

cosχ×ψ+ tan 2 χ+ψ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbGaeq4XdmMaey41aqRaeqiYdKNaey4kaSIaciiDaiaacgga caGGUbWaaWbaaSqabeaacaaIYaaaaOGaeq4XdmMaey4kaSIaeqiYdK haaa@4773@

cosχ×ψ+ tan 2 ( χ+ψ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbGaeq4XdmMaey41aqRaeqiYdKNaey4kaSIaciiDaiaacgga caGGUbWaaWbaaSqabeaacaaIYaaaaOWaaeWaaeaacqaHhpWycqGHRa WkcqaHipqEaiaawIcacaGLPaaaaaa@48FC@

tan ( χ+2×ψ ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiDaiaacg gacaGGUbWaaeWaaeaacqaHhpWycqGHRaWkcaaIYaGaey41aqRaeqiY dKhacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaaa@4274@

sin 2 (χ+ψ)+cosχ+ ψ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacM gacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaaiikaiabeE8aJjabgUca RiabeI8a5jaacMcacqGHRaWkciGGJbGaai4BaiaacohacqaHhpWycq GHRaWkcqaHipqEdaahaaWcbeqaaiaaikdaaaaaaa@4887@

cos( xψ )+ tan 2 x+ψ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbWaaeWaaeaacaWG4bGaeqiYdKhacaGLOaGaayzkaaGaey4k aSIaciiDaiaacggacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaamiEai abgUcaRiabeI8a5baa@4571@

cos( xψ )+ tan 2 ( χ+y ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbWaaeWaaeaacaWG4bGaeqiYdKhacaGLOaGaayzkaaGaey4k aSIaciiDaiaacggacaGGUbWaaWbaaSqabeaacaaIYaaaaOWaaeWaae aacqaHhpWycqGHRaWkcaWG5baacaGLOaGaayzkaaaaaa@46E4@

cosχy+ tan 2 χ+y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbGaeq4XdmMaamyEaiabgUcaRiGacshacaGGHbGaaiOBamaa CaaaleqabaGaaGOmaaaakiabeE8aJjabgUcaRiaadMhaaaa@43BC@

cosχy+ tan 2 ( x+ψ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbGaeq4XdmMaamyEaiabgUcaRiGacshacaGGHbGaaiOBamaa CaaaleqabaGaaGOmaaaakmaabmaabaGaamiEaiabgUcaRiabeI8a5b GaayjkaiaawMcaaaaa@455B@

cosh 2 χ sinh 2 χ=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbGaaiiAamaaCaaaleqabaGaaGOmaaaakiabeE8aJjabgkHi TiGacohacaGGPbGaaiOBaiaacIgadaahaaWcbeqaaiaaikdaaaGccq aHhpWycqGH9aqpcaaIXaaaaa@457C@

sin 2 χ+ cos 2 χ=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacM gacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaeq4XdmMaey4kaSIaci4y aiaac+gacaGGZbWaaWbaaSqabeaacaaIYaaaaOGaeq4XdmMaeyypa0 JaaGymaaaa@4399@

sin 1 χsin χ 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacM gacaGGUbWaaWbaaSqabeaacqGHsislcaaIXaaaaOGaeq4XdmMaeyiy IKRaci4CaiaacMgacaGGUbGaeq4Xdm2aaWbaaSqabeaacqGHsislca aIXaaaaaaa@4490@

cos( χ+ψ )=cosχcosψsinχsinψ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbWaaeWaaeaacqaHhpWycqGHRaWkcqaHipqEaiaawIcacaGL PaaacqGH9aqpciGGJbGaai4BaiaacohacqaHhpWycqGHflY1ciGGJb Gaai4BaiaacohacqaHipqEcqGHsislciGGZbGaaiyAaiaac6gacqaH hpWycqGHflY1ciGGZbGaaiyAaiaac6gacqaHipqEaaa@57A1@

cos2χ= cos 2 χ sin 2 χ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbGaaGOmaiabgwSixlabeE8aJjabg2da9iGacogacaGGVbGa ai4CamaaCaaaleqabaGaaGOmaaaakiabeE8aJjabgkHiTiGacohaca GGPbGaaiOBamaaCaaaleqabaGaaGOmaaaakiabeE8aJbaa@4A79@

9.2 Complex

sinνΑ=sinΑ{ ( 2cosΑ ) ν1 ( ν2 1 ) ( 2cosΑ ) ν3 +( ν3 2 ) ( 2cosΑ ) ν5 } MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacM gacaGGUbGaeqyVd4MaeuyKdeKaeyypa0Jaci4CaiaacMgacaGGUbGa euyKde0aaiWaaqaabeqaamaabmaabaGaaGOmaiGacogacaGGVbGaai 4Caiabfg5abbGaayjkaiaawMcaamaaCaaaleqabaGaeqyVd4MaeyOe I0IaaGymaaaakiabgkHiTmaabmaaeaqabeaacqaH9oGBcqGHsislca aIYaaabaGaaGymaaaacaGLOaGaayzkaaWaaeWaaeaacaaIYaGaci4y aiaac+gacaGGZbGaeuyKdeeacaGLOaGaayzkaaWaaWbaaSqabeaacq aH9oGBcqGHsislcaaIZaaaaaGcbaGaey4kaSYaaeWaaqaabeqaaiab e27aUjabgkHiTiaaiodaaeaacaaIYaaaaiaawIcacaGLPaaadaqada qaaiaaikdaciGGJbGaai4BaiaacohacqqHroqqaiaawIcacaGLPaaa daahaaWcbeqaaiabe27aUjabgkHiTiaaiwdaaaGccqGHsislcqWIVl ctaaGaay5Eaiaaw2haaaaa@7193@

cosνΑ= 1 2 { ( 2cosΑ ) ν ν 1 ( 2cosΑ ) ν2 + ν 2 ( ν3 1 ) ( 2cosΑ ) ν4 ν 3 ( ν4 2 ) ( 2cosΑ ) ν6 + } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbGaeqyVd4MaeuyKdeKaeyypa0ZaaSaaaeaacaaIXaaabaGa aGOmaaaadaGadaabaeqabaWaaeWaaeaacaaIYaGaci4yaiaac+gaca GGZbGaeuyKdeeacaGLOaGaayzkaaWaaWbaaSqabeaacqaH9oGBaaGc cqGHsisldaWcaaqaaiabe27aUbqaaiaaigdaaaWaaeWaaeaacaaIYa Gaci4yaiaac+gacaGGZbGaeuyKdeeacaGLOaGaayzkaaWaaWbaaSqa beaacqaH9oGBcqGHsislcaaIYaaaaOGaey4kaSYaaSaaaeaacqaH9o GBaeaacaaIYaaaamaabmaaeaqabeaacqaH9oGBcqGHsislcaaIZaaa baGaaGymaaaacaGLOaGaayzkaaWaaeWaaeaacaaIYaGaci4yaiaac+ gacaGGZbGaeuyKdeeacaGLOaGaayzkaaWaaWbaaSqabeaacqaH9oGB cqGHsislcaaI0aaaaaGcbaGaeyOeI0YaaSaaaeaacqaH9oGBaeaaca aIZaaaamaabmaaeaqabeaacqaH9oGBcqGHsislcaaI0aaabaGaaGOm aaaacaGLOaGaayzkaaWaaeWaaeaacaaIYaGaci4yaiaac+gacaGGZb GaeuyKdeeacaGLOaGaayzkaaWaaWbaaSqabeaacqaH9oGBcqGHsisl caaI2aaaaOGaey4kaSIaeS47IWeaaiaawUhacaGL9baaaaa@7FB8@

sin 2ν1 Α= ( 1 ) ν1 2 2ν2 { sin( 2ν1 )Α( 2ν1 1 )sin( 2ν3 )Α + ( 1 ) ν1 ( 2ν1 ν1 )sinΑ } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacM gacaGGUbWaaWbaaSqabeaacaaIYaGaeqyVd4MaeyOeI0IaaGymaaaa kiabfg5abjabg2da9maalaaabaWaaeWaaeaacqGHsislcaaIXaaaca GLOaGaayzkaaWaaWbaaSqabeaacqaH9oGBcqGHsislcaaIXaaaaaGc baGaaGOmamaaCaaaleqabaGaaGOmaiabe27aUjabgkHiTiaaikdaaa aaaOWaaiWaaqaabeqaaiGacohacaGGPbGaaiOBamaabmaabaGaaGOm aiabe27aUjabgkHiTiaaigdaaiaawIcacaGLPaaacqqHroqqcqGHsi sldaqadaabaeqabaGaaGOmaiabe27aUjabgkHiTiaaigdaaeaacaaI XaaaaiaawIcacaGLPaaaciGGZbGaaiyAaiaac6gadaqadaqaaiaaik dacqaH9oGBcqGHsislcaaIZaaacaGLOaGaayzkaaGaeuyKdeeabaGa ey4kaSIaeSOjGS0aaeWaaeaacqGHsislcaaIXaaacaGLOaGaayzkaa WaaWbaaSqabeaacqaH9oGBcqGHsislcaaIXaaaaOWaaeWaaqaabeqa aiaaikdacqaH9oGBcqGHsislcaaIXaaabaGaeqyVd4MaeyOeI0IaaG ymaaaacaGLOaGaayzkaaGaci4CaiaacMgacaGGUbGaeuyKdeeaaiaa wUhacaGL9baaaaa@7EC8@

cos 2ν1 Α= 1 2 2ν2 { cos( 2ν1 )Α+( 2ν1 1 )cos( 2ν3 )Α ++( 2ν1 ν1 )cosΑ } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbWaaWbaaSqabeaacaaIYaGaeqyVd4MaeyOeI0IaaGymaaaa kiabfg5abjabg2da9maalaaabaGaaGymaaqaaiaaikdadaahaaWcbe qaaiaaikdacqaH9oGBcqGHsislcaaIYaaaaaaakmaacmaaeaqabeaa ciGGJbGaai4BaiaacohadaqadaqaaiaaikdacqaH9oGBcqGHsislca aIXaaacaGLOaGaayzkaaGaeuyKdeKaey4kaSYaaeWaaqaabeqaaiaa ikdacqaH9oGBcqGHsislcaaIXaaabaGaaGymaaaacaGLOaGaayzkaa Gaci4yaiaac+gacaGGZbWaaeWaaeaacaaIYaGaeqyVd4MaeyOeI0Ia aG4maaGaayjkaiaawMcaaiabfg5abbqaaiabgUcaRiabl+UimjabgU caRmaabmaaeaqabeaacaaIYaGaeqyVd4MaeyOeI0IaaGymaaqaaiab e27aUjabgkHiTiaaigdaaaGaayjkaiaawMcaaiGacogacaGGVbGaai 4Caiabfg5abbaacaGL7bGaayzFaaaaaa@7382@

coshχ= e χ + e χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbGaaiiAaiabeE8aJjabg2da9maalaaabaGaamyzamaaCaaa leqabaGaeq4XdmgaaOGaey4kaSIaamyzamaaCaaaleqabaGaeyOeI0 Iaeq4XdmgaaaGcbaGaaGOmaaaaaaa@44BE@

10. Mix of Greek & English

( x+ψ+z ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WG4bGaey4kaSIaeqiYdKNaey4kaSIaamOEaaGaayjkaiaawMcaamaa CaaaleqabaGaaGOmaaaaaaa@3DF7@

( χ+y ) 2 +z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aHhpWycqGHRaWkcaWG5baacaGLOaGaayzkaaWaaWbaaSqabeaacaaI YaaaaOGaey4kaSIaamOEaaaa@3DEB@

x+ ( ψ+ζ ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgU caRmaabmaabaGaeqiYdKNaey4kaSIaeqOTdOhacaGLOaGaayzkaaWa aWbaaSqabeaacaaIYaaaaaaa@3EB5@

x+ψ+ z 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEaiabgU caRiabeI8a5jabgUcaRiaadQhadaahaaWcbeqaaiaaikdaaaaaaa@3C6E@

b c +δε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaada WcaaqaaiaadkgaaeaacaWGJbaaaaWcbeaakiabgUcaRiabes7aKjab gkHiTiabew7aLbaa@3D16@

β c+d ε MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaada Wcaaqaaiabek7aIbqaaiaadogacqGHRaWkcaWGKbaaaaWcbeaakiab gkHiTiabew7aLbaa@3D14@

β c e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaada Wcaaqaaiabek7aIbqaaiaadogaaaGaeyOeI0IaamyzaaWcbeaaaaa@3A82@

cos( xψ )+ tan 2 x+ψ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbWaaeWaaeaacaWG4bGaeqiYdKhacaGLOaGaayzkaaGaey4k aSIaciiDaiaacggacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaamiEai abgUcaRiabeI8a5baa@4571@

cos( xψ )+ tan 2 ( χ+y ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbWaaeWaaeaacaWG4bGaeqiYdKhacaGLOaGaayzkaaGaey4k aSIaciiDaiaacggacaGGUbWaaWbaaSqabeaacaaIYaaaaOWaaeWaae aacqaHhpWycqGHRaWkcaWG5baacaGLOaGaayzkaaaaaa@46E4@

cosχy+ tan 2 χ+y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbGaeq4XdmMaamyEaiabgUcaRiGacshacaGGHbGaaiOBamaa CaaaleqabaGaaGOmaaaakiabeE8aJjabgUcaRiaadMhaaaa@43BC@

cosχy+ tan 2 ( x+ψ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbGaeq4XdmMaamyEaiabgUcaRiGacshacaGGHbGaaiOBamaa CaaaleqabaGaaGOmaaaakmaabmaabaGaamiEaiabgUcaRiabeI8a5b GaayjkaiaawMcaaaaa@455B@

11. Text and Math

11.1 Greek Text

Πολλαπλασιασμός πολυωνύμων

Μαθαίνω να πολλαπλασιάζω

Μονώνυμο με πολυώνυμο

Πολυώνυμο με πολυώνυμο

Δραστηριότητα

Να γράψετε το γινόμενο α×( β+γ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey 41aq7aaeWaaeaacqaHYoGycqGHRaWkcqaHZoWzaiaawIcacaGLPaaa aaa@3F60@ σύμφωνα με την επιμεριστική ιδιότητα και με ανάλογο τρόπο να βρείτε την παράσταση 3× χ 2 ×( 2 χ 3 +6χ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiabgE na0kabeE8aJnaaCaaaleqabaGaaGOmaaaakiabgEna0oaabmaabaGa aGOmaiabeE8aJnaaCaaaleqabaGaaG4maaaakiabgUcaRiaaiAdacq aHhpWyaiaawIcacaGLPaaaaaa@45D5@

Να γράψετε το γινόμενο ( α+β )×( γ+δ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aHXoqycqGHRaWkcqaHYoGyaiaawIcacaGLPaaacqGHxdaTdaqadaqa aiabeo7aNjabgUcaRiabes7aKbGaayjkaiaawMcaaaaa@4370@    σύμφωνα με την επιμεριστική ιδιότητα και με ανάλογο τρόπο να βρείτε την παράσταση ( 3 χ 2 ψ+2ψ )×( 2 χ 2 +5 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIZaGaeq4Xdm2aaWbaaSqabeaacaaIYaaaaOGaeqiYdKNaey4kaSIa aGOmaiabeI8a5bGaayjkaiaawMcaaiabgEna0oaabmaabaGaaGOmai abeE8aJnaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaiwdaaiaawIca caGLPaaaaaa@48C8@

Πολλαπλασιασμός μονωνύμου με πολυώνυμο

Την αλγεβρική παράσταση 3 χ 2 ×( 2 χ 3 +6χ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiabeE 8aJnaaCaaaleqabaGaaGOmaaaakiabgEna0oaabmaabaGaaGOmaiab eE8aJnaaCaaaleqabaGaaG4maaaakiabgUcaRiaaiAdacqaHhpWyai aawIcacaGLPaaaaaa@43BE@  που είναι γινόμενο του μονωνύμου 3 χ 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiabeE 8aJnaaCaaaleqabaGaaGOmaaaaaaa@3954@    με το πολυώνυμο 2 χ 3 +6χ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiabeE 8aJnaaCaaaleqabaGaaG4maaaakiabgUcaRiaaiAdacqaHhpWyaaa@3CB7@   σύμφωνα με την επιμεριστική ιδιότητα μπορούμε να την γράψουμε

3 χ 2 ×( 2 χ 3 +6χ )=3 χ 2 ×2 χ 3 +3 χ 2 ×6χ=6 χ 5 +18 χ 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiabeE 8aJnaaCaaaleqabaGaaGOmaaaakiabgEna0oaabmaabaGaaGOmaiab eE8aJnaaCaaaleqabaGaaG4maaaakiabgUcaRiaaiAdacqaHhpWyai aawIcacaGLPaaacqGH9aqpcaaIZaGaeq4Xdm2aaWbaaSqabeaacaaI YaaaaOGaey41aqRaaGOmaiabeE8aJnaaCaaaleqabaGaaG4maaaaki abgUcaRiaaiodacqaHhpWydaahaaWcbeqaaiaaikdaaaGccqGHxdaT caaI2aGaeq4XdmMaeyypa0JaaGOnaiabeE8aJnaaCaaaleqabaGaaG ynaaaakiabgUcaRiaaigdacaaI4aGaeq4Xdm2aaWbaaSqabeaacaaI Zaaaaaaa@5FF3@

Διαπιστώνουμε ότι για να πολλαπλασιάσουμε μονώνυμο με πολυώνυμο, πολλαπλασιάζουμε το μονώνυμο με κάθε όρο του πολυωνύμου και προσθέτουμε τα γινόμενα που προκύπτουν.

Πολλαπλασιασμός πολυωνύμου με πολυώνυμο και πράσθεση-αφαίρεση πολυωνύμων

Την αλγεβρική παράσταση ( 3 χ 2 ψ+2ψ )×( 2 χ 2 +5 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIZaGaeq4Xdm2aaWbaaSqabeaacaaIYaaaaOGaeqiYdKNaey4kaSIa aGOmaiabeI8a5bGaayjkaiaawMcaaiabgEna0oaabmaabaGaaGOmai abeE8aJnaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaiwdaaiaawIca caGLPaaaaaa@48C8@  που είναι γινόμενο του πολυωνύμου 3 χ 2 ψ+2ψ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiabeE 8aJnaaCaaaleqabaGaaGOmaaaakiabeI8a5jabgUcaRiaaikdacqaH ipqEaaa@3E98@  με το πολυώνυμο 2 χ 2 +5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiabeE 8aJnaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaiwdaaaa@3AFE@ , σύμφωνα με την επιμεριστική ιδιότητα μπορούμε να τη γράψουμε

( 3 χ 2 ψ+2ψ )×( 2 χ 2 +5 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIZaGaeq4Xdm2aaWbaaSqabeaacaaIYaaaaOGaeqiYdKNaey4kaSIa aGOmaiabeI8a5bGaayjkaiaawMcaaiabgEna0oaabmaabaGaaGOmai abeE8aJnaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaiwdaaiaawIca caGLPaaaaaa@48C8@ =3 χ 2 ψ×2 χ 2 +3 χ 2 ψ×5+2ψ×2 χ 2 +2ψ×5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG 4maiabeE8aJnaaCaaaleqabaGaaGOmaaaakiabeI8a5jabgEna0kaa ikdacqaHhpWydaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIZaGaeq 4Xdm2aaWbaaSqabeaacaaIYaaaaOGaeqiYdKNaey41aqRaaGynaiab gUcaRiaaikdacqaHipqEcqGHxdaTcaaIYaGaeq4Xdm2aaWbaaSqabe aacaaIYaaaaOGaey4kaSIaaGOmaiabeI8a5jabgEna0kaaiwdaaaa@59C7@

=6 χ 4 ψ+15 χ 2 ψ+4 χ 2 ψ+10ψ=6 χ 4 ψ+19 χ 2 ψ+10ψ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG OnaiabeE8aJnaaCaaaleqabaGaaGinaaaakiabeI8a5jabgUcaRiaa igdacaaI1aGaeq4Xdm2aaWbaaSqabeaacaaIYaaaaOGaeqiYdKNaey 4kaSIaaGinaiabeE8aJnaaCaaaleqabaGaaGOmaaaakiabeI8a5jab gUcaRiaaigdacaaIWaGaeqiYdKNaeyypa0JaaGOnaiabeE8aJnaaCa aaleqabaGaaGinaaaakiabeI8a5jabgUcaRiaaigdacaaI5aGaeq4X dm2aaWbaaSqabeaacaaIYaaaaOGaeqiYdKNaey4kaSIaaGymaiaaic dacqaHipqEaaa@5E85@

Διαπιστώνουμε ότι για να πολλαπλασιάσουμε πολυώνυμο  με πολυώνυμο, πολλαπλασιάζουμε κάθε όρο του ενός πολυωνύμου με κάθε όρο του άλλου  πολυωνύμου και προσθέτουμε τα γινόμενα που προκύπτουν.

Όταν κάνουμε πολλαπλασιασμό μονωνύμου με πολυώνυμο ή δύο πολυωνύμων, λέμε ότι αναπτύσσουμε τα γινόμενα αυτά και το αποτέλεσμα ονομάζεται ανάπτυγμα του γινομένου.

Τριγωνομετρικοί αριθμοί της γωνίας 2α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiabeg 7aHbaa@3852@

Οι τύποι που εκφρρζουν τους τριγωνομετρικούς αριθμούς αυτής της γωνίας ως συνάρτηση των τριγωνομετρικών αριθμών της γωνίας α, είναι ειδικές περιπτώσεις των τύπων της προηγούμενης παραγράφου. Συγκεκριμένα, αν στους τύπους του  sin( α+β ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacM gacaGGUbWaaeWaaeaacqaHXoqycqGHRaWkcqaHYoGyaiaawIcacaGL Paaaaaa@3E7A@ , του cos( α+β ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbWaaeWaaeaacqaHXoqycqGHRaWkcqaHYoGyaiaawIcacaGL Paaaaaa@3E75@  και της tan( α+β ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiDaiaacg gacaGGUbWaaeWaaeaacqaHXoqycqGHRaWkcqaHYoGyaiaawIcacaGL Paaaaaa@3E73@  αντικαταστήσουμε το β με το α, έχουμε

sin( 2α )=sin( α+α )=sin( α )×cos( α )+cos( α )×sin( α )=2×sin( α )×cos( a ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacM gacaGGUbWaaeWaaeaacaaIYaGaeqySdegacaGLOaGaayzkaaGaeyyp a0Jaci4CaiaacMgacaGGUbWaaeWaaeaacqaHXoqycqGHRaWkcqaHXo qyaiaawIcacaGLPaaacqGH9aqpciGGZbGaaiyAaiaac6gadaqadaqa aiabeg7aHbGaayjkaiaawMcaaiabgEna0kGacogacaGGVbGaai4Cam aabmaabaGaeqySdegacaGLOaGaayzkaaGaey4kaSIaci4yaiaac+ga caGGZbWaaeWaaeaacqaHXoqyaiaawIcacaGLPaaacqGHxdaTciGGZb GaaiyAaiaac6gadaqadaqaaiabeg7aHbGaayjkaiaawMcaaiabg2da 9iaaikdacqGHxdaTciGGZbGaaiyAaiaac6gadaqadaqaaiabeg7aHb GaayjkaiaawMcaaiabgEna0kGacogacaGGVbGaai4CamaabmaabaGa amyyaaGaayjkaiaawMcaaaaa@7578@ Επομένως:  sin( 2α )=2×sin( α )×cos( α ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacM gacaGGUbWaaeWaaeaacaaIYaGaeqySdegacaGLOaGaayzkaaGaeyyp a0JaaGOmaiabgEna0kGacohacaGGPbGaaiOBamaabmaabaGaeqySde gacaGLOaGaayzkaaGaey41aqRaci4yaiaac+gacaGGZbWaaeWaaeaa cqaHXoqyaiaawIcacaGLPaaaaaa@4E9E@

cos( 2α )=cos( α+α )=cos( α )×cos( α )sin( α )×sin( α )= cos 2 α sin 2 α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbWaaeWaaeaacaaIYaGaeqySdegacaGLOaGaayzkaaGaeyyp a0Jaci4yaiaac+gacaGGZbWaaeWaaeaacqaHXoqycqGHRaWkcqaHXo qyaiaawIcacaGLPaaacqGH9aqpciGGJbGaai4Baiaacohadaqadaqa aiabeg7aHbGaayjkaiaawMcaaiabgEna0kGacogacaGGVbGaai4Cam aabmaabaGaeqySdegacaGLOaGaayzkaaGaeyOeI0Iaci4CaiaacMga caGGUbWaaeWaaeaacqaHXoqyaiaawIcacaGLPaaacqGHxdaTciGGZb GaaiyAaiaac6gadaqadaqaaiabeg7aHbGaayjkaiaawMcaaiabg2da 9iGacogacaGGVbGaai4CamaaCaaaleqabaGaaGOmaaaakiabeg7aHj abgkHiTiGacohacaGGPbGaaiOBamaaCaaaleqabaGaaGOmaaaakiab eg7aHbaa@7109@ = cos 2 α(1 cos 2 α)=2 cos 2 α1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaci 4yaiaac+gacaGGZbWaaWbaaSqabeaacaaIYaaaaOGaeqySdeMaeyOe I0IaaiikaiaaigdacqGHsislciGGJbGaai4BaiaacohadaahaaWcbe qaaiaaikdaaaGccqaHXoqycaGGPaGaeyypa0JaaGOmaiGacogacaGG VbGaai4CamaaCaaaleqabaGaaGOmaaaakiabeg7aHjabgkHiTiaaig daaaa@4E84@

=(1 sin 2 α) sin 2 α=12 sin 2 α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaai ikaiaaigdacqGHsislciGGZbGaaiyAaiaac6gadaahaaWcbeqaaiaa ikdaaaGccqaHXoqycaGGPaGaeyOeI0Iaci4CaiaacMgacaGGUbWaaW baaSqabeaacaaIYaaaaOGaeqySdeMaeyypa0JaaGymaiabgkHiTiaa ikdaciGGZbGaaiyAaiaac6gadaahaaWcbeqaaiaaikdaaaGccqaHXo qyaaa@4E93@

Επίσης: cos( 2α )= cos 2 α sin 2 α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbWaaeWaaeaacaaIYaGaeqySdegacaGLOaGaayzkaaGaeyyp a0Jaci4yaiaac+gacaGGZbWaaWbaaSqabeaacaaIYaaaaOGaeqySde MaeyOeI0Iaci4CaiaacMgacaGGUbWaaWbaaSqabeaacaaIYaaaaOGa eqySdegaaa@4970@

Επομένως:  cos( 2α )= cos 2 α sin 2 α=2 cos 2 α1=12 sin 2 α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbWaaeWaaeaacaaIYaGaeqySdegacaGLOaGaayzkaaGaeyyp a0Jaci4yaiaac+gacaGGZbWaaWbaaSqabeaacaaIYaaaaOGaeqySde MaeyOeI0Iaci4CaiaacMgacaGGUbWaaWbaaSqabeaacaaIYaaaaOGa eqySdeMaeyypa0JaaGOmaiGacogacaGGVbGaai4CamaaCaaaleqaba GaaGOmaaaakiabeg7aHjabgkHiTiaaigdacqGH9aqpcaaIXaGaeyOe I0IaaGOmaiGacohacaGGPbGaaiOBamaaCaaaleqabaGaaGOmaaaaki abeg7aHbaa@5B13@

tan(2α)= tanα+tanα 1tanαtanα = 2tanα 1 tan 2 α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiDaiaacg gacaGGUbGaaiikaiaaikdacqaHXoqycaGGPaGaeyypa0ZaaSaaaeaa ciGG0bGaaiyyaiaac6gacqaHXoqycqGHRaWkciGG0bGaaiyyaiaac6 gacqaHXoqyaeaacaaIXaGaeyOeI0IaciiDaiaacggacaGGUbGaeqyS deMaciiDaiaacggacaGGUbGaeqySdegaaiabg2da9maalaaabaGaaG OmaiGacshacaGGHbGaaiOBaiabeg7aHbqaaiaaigdacqGHsislciGG 0bGaaiyyaiaac6gadaahaaWcbeqaaiaaikdaaaGccqaHXoqyaaaaaa@5F29@ , επομένως tan(2α)= 2tanα 1 tan 2 α MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiDaiaacg gacaGGUbGaaiikaiaaikdacqaHXoqycaGGPaGaeyypa0ZaaSaaaeaa caaIYaGaciiDaiaacggacaGGUbGaeqySdegabaGaaGymaiabgkHiTi GacshacaGGHbGaaiOBamaaCaaaleqabaGaaGOmaaaakiabeg7aHbaa aaa@49C9@

cos(2α)=2 cos 2 α11+cos( 2α )=2 cos 2 α cos 2 α= 1+cos2α 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbGaaiikaiaaikdacqaHXoqycaGGPaGaeyypa0JaaGOmaiGa cogacaGGVbGaai4CamaaCaaaleqabaGaaGOmaaaakiabeg7aHjabgk HiTiaaigdacqGHuhY2caaIXaGaey4kaSIaci4yaiaac+gacaGGZbWa aeWaaeaacaaIYaGaeqySdegacaGLOaGaayzkaaGaeyypa0JaaGOmai GacogacaGGVbGaai4CamaaCaaaleqabaGaaGOmaaaakiabeg7aHjab gsDiBlGacogacaGGVbGaai4CamaaCaaaleqabaGaaGOmaaaakiabeg 7aHjabg2da9maalaaabaGaaGymaiabgUcaRiGacogacaGGVbGaai4C aiaaikdacqaHXoqyaeaacaaIYaaaaaaa@6782@

cos( 2α )=12 sin 2 α2 sin 2 α=1cos(2α)si n 2 α= 1cos( 2α ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbWaaeWaaeaacaaIYaGaeqySdegacaGLOaGaayzkaaGaeyyp a0JaaGymaiabgkHiTiaaikdaciGGZbGaaiyAaiaac6gadaahaaWcbe qaaiaaikdaaaGccqaHXoqycqGHuhY2caaIYaGaci4CaiaacMgacaGG UbWaaWbaaSqabeaacaaIYaaaaOGaeqySdeMaeyypa0JaaGymaiabgk HiTiGacogacaGGVbGaai4CaiaacIcacaaIYaGaeqySdeMaaiykaiab gsDiBlaacohacaGGPbGaaiOBamaaCaaaleqabaGaaGOmaaaakiabeg 7aHjabg2da9maalaaabaGaaGymaiabgkHiTiGacogacaGGVbGaai4C amaabmaabaGaaGOmaiabeg7aHbGaayjkaiaawMcaaaqaaiaaikdaaa aaaa@692E@

tan 2 α= sin 2 α cos 2 α = 1cos(2α) 2 1+cos( 2α ) 2 = 1cos(2α) 1+cos(2α) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiDaiaacg gacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaeqySdeMaeyypa0ZaaSaa aeaaciGGZbGaaiyAaiaac6gadaahaaWcbeqaaiaaikdaaaGccqaHXo qyaeaaciGGJbGaai4BaiaacohadaahaaWcbeqaaiaaikdaaaGccqaH XoqyaaGaeyypa0ZaaSaaaeaadaWcaaqaaiaaigdacqGHsislciGGJb Gaai4BaiaacohacaGGOaGaaGOmaiabeg7aHjaacMcaaeaacaaIYaaa aaqaamaalaaabaGaaGymaiabgUcaRiGacogacaGGVbGaai4Camaabm aabaGaaGOmaiabeg7aHbGaayjkaiaawMcaaaqaaiaaikdaaaaaaiab g2da9maalaaabaGaaGymaiabgkHiTiGacogacaGGVbGaai4CaiaacI cacaaIYaGaeqySdeMaaiykaaqaaiaaigdacqGHRaWkciGGJbGaai4B aiaacohacaGGOaGaaGOmaiabeg7aHjaacMcaaaaaaa@6BD9@

Με τη βοήθεια των παραπάνω τύπων μπορούμε να υπολογίσουμε τους τριγωνομετρικούς αριθμούς του μισού μιας γωνίας , αν γνωρίζουμε τους τριγωνομετρικούς αριθμούς της γωνίας αυτής. Για παράδειγμα οι τριγωνομετρικοί αριθμοί της γωνίας 22.5°= 45° 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaaik dacaGGUaGaaGynaiabgclaWkabg2da9maalaaabaGaaGinaiaaiwda cqGHWcaSaeaacaaIYaaaaaaa@4007@   υπολογίζονται ως εξής:

sin 2 ( 22.5° )= 1cos( 45° ) 2 = 1 2 2 2 = 2 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacM gacaGGUbWaaWbaaSqabeaacaaIYaaaaOWaaeWaaeaacaaIYaGaaGOm aiaac6cacaaI1aGaeyiSaalacaGLOaGaayzkaaGaeyypa0ZaaSaaae aacaaIXaGaeyOeI0Iaci4yaiaac+gacaGGZbWaaeWaaeaacaaI0aGa aGynaiabgclaWcGaayjkaiaawMcaaaqaaiaaikdaaaGaeyypa0ZaaS aaaeaacaaIXaGaeyOeI0YaaSaaaeaadaGcaaqaaiaaikdaaSqabaaa keaacaaIYaaaaaqaaiaaikdaaaGaeyypa0ZaaSaaaeaacaaIYaGaey OeI0YaaOaaaeaacaaIYaaaleqaaaGcbaGaaGOmaaaaaaa@54E2@ sin( 22.5° )= 2 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taci 4CaiaacMgacaGGUbWaaeWaaeaacaaIYaGaaGOmaiaac6cacaaI1aGa eyiSaalacaGLOaGaayzkaaGaeyypa0ZaaSaaaeaadaGcaaqaaiaaik dacqGHsisldaGcaaqaaiaaikdaaSqabaaabeaaaOqaaiaaikdaaaaa aa@45F6@

cos 2 (22.5°)= 1+cos( 45° ) 2 = 1+ 2 2 2 = 2+ 2 2 cos(22.5°)= 2+ 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbWaaWbaaSqabeaacaaIYaaaaOGaaiikaiaaikdacaaIYaGa aiOlaiaaiwdacqGHWcaScaGGPaGaeyypa0ZaaSaaaeaacaaIXaGaey 4kaSIaci4yaiaac+gacaGGZbWaaeWaaeaacaaI0aGaaGynaiabgcla WcGaayjkaiaawMcaaaqaaiaaikdaaaGaeyypa0ZaaSaaaeaacaaIXa Gaey4kaSYaaSaaaeaadaGcaaqaaiaaikdaaSqabaaakeaacaaIYaaa aaqaaiaaikdaaaGaeyypa0ZaaSaaaeaacaaIYaGaey4kaSYaaOaaae aacaaIYaaaleqaaaGcbaGaaGOmaaaacqGHshI3ciGGJbGaai4Baiaa cohacaGGOaGaaGOmaiaaikdacaGGUaGaaGynaiabgclaWkaacMcacq GH9aqpdaWcaaqaamaakaaabaGaaGOmaiabgUcaRmaakaaabaGaaGOm aaWcbeaaaeqaaaGcbaGaaGOmaaaaaaa@644B@

Επομένως: tan(22.5°)= 2 2 2+ 2 = 2 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiDaiaacg gacaGGUbGaaiikaiaaikdacaaIYaGaaiOlaiaaiwdacqGHWcaScaGG PaGaeyypa0ZaaSaaaeaadaGcaaqaaiaaikdacqGHsisldaGcaaqaai aaikdaaSqabaaabeaaaOqaamaakaaabaGaaGOmaiabgUcaRmaakaaa baGaaGOmaaWcbeaaaeqaaaaakiabg2da9maakaaabaGaaGOmaaWcbe aakiabgkHiTiaaigdaaaa@48C4@   και cot(22.5°)= 2+ 2 2 2 = 2 +1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGG0bGaaiikaiaaikdacaaIYaGaaiOlaiaaiwdacqGHWcaScaGG PaGaeyypa0ZaaSaaaeaadaGcaaqaaiaaikdacqGHRaWkdaGcaaqaai aaikdaaSqabaaabeaaaOqaamaakaaabaGaaGOmaiabgkHiTmaakaaa baGaaGOmaaWcbeaaaeqaaaaakiabg2da9maakaaabaGaaGOmaaWcbe aakiabgUcaRiaaigdaaaa@48BC@

Εκτίμηση άγνωστων συναρτήσεων πυκνότητας πιθανότητας και εκτίμηση παραμέτρων με τη μέθοδο μέγιστης πιθανοφάνειας

Ας θεωρήσουμε ένα πρόβλημα Μ κλάσεων, με διανύσματα χαρακτηριστικών κατανεμημένα σύμφωνα με τις p(χ| ω ι ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiaacI cacqaHhpWydaabbaqaaiabeM8a3naaBaaaleaacqaH5oqAaeqaaOGa aiykaaGaay5bSdaaaa@3F42@ από 1 ως Μ. Υποθέτουμε ότι αυτές οι συναρτήσεις πιθανοφάνειας δίνονται σε παραμετρική μορφή και ότι οι αντίστοιχες παράμετροι σχηματίζουν διανύσματα θ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUde3aaS baaSqaaiaadMgaaeqaaaaa@38C7@ που είναι άγνωστα. Προκειμένου να δείξουμε την εξάρτηση των συναρτήσεων πυκνότητας πιθανότητας από τα διανύσματα αυτά γράφουμε p(χ| ω ι ;θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiaacI cacqaHhpWydaabbaqaaiabeM8a3naaBaaaleaacqaH5oqAaeqaaOGa ai4oaiabeI7aXjaacMcaaiaawEa7aaaa@41B7@ . Ο στόχος μας είναι να εκτιμήσουμε τις άγνωστες παραμέτρους χρησιμοποιώντας ένα σύνολο από διαθέσιμα διανύσματα χαρακτηριστικών από κάθε κλήση.

Ας υποθέσουμε ότι χ 1 ,, χ Ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaS baaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacqaHhpWydaWg aaWcbaGaeuyNd4eabeaaaaa@3E7D@   είναι τυχαία δείγματα που προέκυψαν από τη συνάρτηση πυκνότητας πιθανότητας. Σχηματίζουμε την από κοινού συνάρτηση πυκνότητας πιθανότητας p(Χ;θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiaacI cacqqHNoWqcaGG7aGaeqiUdeNaaiykaaaa@3C32@ . Υποθέτωντας στατιστική ανεξαρτησία μεταξύ των δειγμάτων, έχουμε

p(Χ;θ)=p( χ 1 , χ 2 ,, χ Ν ;θ)= κ=1 Ν p( χ κ ;θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiaacI cacqqHNoWqcaGG7aGaeqiUdeNaaiykaiabg2da9iaacchacaGGOaGa eq4Xdm2aaSbaaSqaaiaaigdaaeqaaOGaaiilaiabeE8aJnaaBaaale aacaaIYaaabeaakiaacYcacqWIMaYscaGGSaGaeq4Xdm2aaSbaaSqa aiabf25aobqabaGccaGG7aGaeqiUdeNaaiykaiabg2da9maarahaba GaamiCaiaacIcacqaHhpWydaWgaaWcbaGaeqOUdSgabeaakiaacUda cqaH4oqCcaGGPaaaleaacqaH6oWAcqGH9aqpcaaIXaaabaGaeuyNd4 eaniabg+Givdaaaa@5E68@

Αυτή είναι μια συνάρτηση του θ και είναι γνωστή ως συνάρτηση πιθανοφάνειας του θ ως προς Χ. Η μέθοδος μέγιστης πιθανοφάνειας εκτιμά το θ ώστε η συνάρτηση πιθανοφάνειας να λαμβάνει την μέγιστη τιμή της, δηλαδή

θ ML =arg max θ κ=1 Ν p( χ κ ;θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUde3aaS baaSqaaiaad2eacaWGmbaabeaakiabg2da9iGacggacaGGYbGaai4z aiGac2gacaGGHbGaaiiEamaaBaaaleaacqaH4oqCaeqaaOWaaebCae aacaWGWbGaaiikaiabeE8aJnaaBaaaleaacqaH6oWAaeqaaOGaai4o aiabeI7aXjaacMcaaSqaaiabeQ7aRjabg2da9iaaigdaaeaacqqHDo Gta0Gaey4dIunaaaa@5193@

Μια αναγκαστική συνθήκη που πρέπει να πληρεί αυτό το θ είναι να είναι μέγιστο, άρα να μηδενίζει την παράγωγο της συνάρτησης πιθανοφάνειας ως προς θ, δηλαδή:

κ=1 Ν p( χ κ ;θ) θ =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHciITdaqeWbqaaiaadchacaGGOaGaeq4Xdm2aaSbaaSqaaiabeQ7a RbqabaGccaGG7aGaeqiUdeNaaiykaaWcbaGaeqOUdSMaeyypa0JaaG ymaaqaaiabf25aobqdcqGHpis1aaGcbaGaeyOaIyRaeqiUdehaaiab g2da9iaaicdaaaa@4BD2@

Εξαιτίας της γνησίως αύξουσας μονοτονίας της λογαριθμικής συνάρτησης, ορίζουμε τη λογαριθμική συνάρτηση πιθανοφάνειας ως:

L(θ)=ln κ=1 Ν p( χ κ ;θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaiaacI cacqaH4oqCcaGGPaGaeyypa0JaciiBaiaac6gadaqeWbqaaiaadcha caGGOaGaeq4Xdm2aaSbaaSqaaiabeQ7aRbqabaGccaGG7aGaeqiUde NaaiykaaWcbaGaeqOUdSMaeyypa0JaaGymaaqaaiabf25aobqdcqGH pis1aaaa@4C40@

Επομένως τώρα έχουμε το εξής:

L(θ) θ = κ=1 Ν lnp( χ κ ;θ) θ = κ=1 Ν 1 p( χ κ ;θ) × p( χ κ ;θ) θ =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHciITcaWGmbGaaiikaiabeI7aXjaacMcaaeaacqGHciITcqaH4oqC aaGaeyypa0ZaaabCaeaadaWcaaqaaiabgkGi2kGacYgacaGGUbGaam iCaiaacIcacqaHhpWydaWgaaWcbaGaeqOUdSgabeaakiaacUdacqaH 4oqCcaGGPaaabaGaeyOaIyRaeqiUdehaaaWcbaGaeqOUdSMaeyypa0 JaaGymaaqaaiabf25aobqdcqGHris5aOGaeyypa0ZaaabCaeaadaWc aaqaaiaaigdaaeaacaWGWbGaaiikaiabeE8aJnaaBaaaleaacqaH6o WAaeqaaOGaai4oaiabeI7aXjaacMcaaaGaey41aq7aaSaaaeaacqGH ciITcaWGWbGaaiikaiabeE8aJnaaBaaaleaacqaH6oWAaeqaaOGaai 4oaiabeI7aXjaacMcaaeaacqGHciITcqaH4oqCaaaaleaacqaH6oWA cqGH9aqpcaaIXaaabaGaeuyNd4eaniabggHiLdGccqGH9aqpcaaIWa aaaa@77B5@

11.2 English Text

Multiplying Polynomials

Learn how to multiply

  • Monomials with Polynomials
  • Polynomials with Polynomials
Activity

Use the distributive law to write the product a×( b+c ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey 41aq7aaeWaaeaacqaHYoGycqGHRaWkcqaHZoWzaiaawIcacaGLPaaa aaa@3F60@ and find the expression 3× x 2 ×( 2 x 3 +6x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiabgE na0kabeE8aJnaaCaaaleqabaGaaGOmaaaakiabgEna0oaabmaabaGa aGOmaiabeE8aJnaaCaaaleqabaGaaG4maaaakiabgUcaRiaaiAdacq aHhpWyaiaawIcacaGLPaaaaaa@45D5@ accordingly

Use the distributive law to write the product ( a+b )×( c+d ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaacq aHXoqycqGHRaWkcqaHYoGyaiaawIcacaGLPaaacqGHxdaTdaqadaqa aiabeo7aNjabgUcaRiabes7aKbGaayjkaiaawMcaaaaa@4370@    and find the expression ( 3 x 2 ψ+2ψ )×( 2 x 2 +5 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIZaGaeq4Xdm2aaWbaaSqabeaacaaIYaaaaOGaeqiYdKNaey4kaSIa aGOmaiabeI8a5bGaayjkaiaawMcaaiabgEna0oaabmaabaGaaGOmai abeE8aJnaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaiwdaaiaawIca caGLPaaaaaa@48C8@ accordingly

Small experiment
Multiplying monomials with polynomials

Following the distributive law, we can write the algebraic expression 3 x 2 ×( 2 x 3 +6x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiabeE 8aJnaaCaaaleqabaGaaGOmaaaakiabgEna0oaabmaabaGaaGOmaiab eE8aJnaaCaaaleqabaGaaG4maaaakiabgUcaRiaaiAdacqaHhpWyai aawIcacaGLPaaaaaa@43BE@  as a product of the monomial 3 x 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiabeE 8aJnaaCaaaleqabaGaaGOmaaaaaaa@3954@    with the polynomial 2 x 3 +6x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiabeE 8aJnaaCaaaleqabaGaaG4maaaakiabgUcaRiaaiAdacqaHhpWyaaa@3CB7@   as follows

3 x 2 ×( 2 x 3 +6x )=3 x 2 ×2 x 3 +3 x 2 ×6x=6 x 5 +18 x 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiabeE 8aJnaaCaaaleqabaGaaGOmaaaakiabgEna0oaabmaabaGaaGOmaiab eE8aJnaaCaaaleqabaGaaG4maaaakiabgUcaRiaaiAdacqaHhpWyai aawIcacaGLPaaacqGH9aqpcaaIZaGaeq4Xdm2aaWbaaSqabeaacaaI YaaaaOGaey41aqRaaGOmaiabeE8aJnaaCaaaleqabaGaaG4maaaaki abgUcaRiaaiodacqaHhpWydaahaaWcbeqaaiaaikdaaaGccqGHxdaT caaI2aGaeq4XdmMaeyypa0JaaGOnaiabeE8aJnaaCaaaleqabaGaaG ynaaaakiabgUcaRiaaigdacaaI4aGaeq4Xdm2aaWbaaSqabeaacaaI Zaaaaaaa@5FF3@

We conclude that in order to multiply a monomial with a polynomial, we have to multiply the monomial with each term of the polynomial and then add the resulting products.

Multiplying polynomials with polynomials and polynomial addition-subtraction

Following the distributive law, we can write the algebraic expression ( 3 x 2 y+2y )×( 2 x 2 +5 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIZaGaeq4Xdm2aaWbaaSqabeaacaaIYaaaaOGaeqiYdKNaey4kaSIa aGOmaiabeI8a5bGaayjkaiaawMcaaiabgEna0oaabmaabaGaaGOmai abeE8aJnaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaiwdaaiaawIca caGLPaaaaaa@48C8@  as a product of the polynomial 3 x 2 y+2y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiabeE 8aJnaaCaaaleqabaGaaGOmaaaakiabeI8a5jabgUcaRiaaikdacqaH ipqEaaa@3E98@  with the polynomial 2 x 2 +5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiabeE 8aJnaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaiwdaaaa@3AFE@ , as

( 3 x 2 y+2y )×( 2 x 2 +5 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIZaGaeq4Xdm2aaWbaaSqabeaacaaIYaaaaOGaeqiYdKNaey4kaSIa aGOmaiabeI8a5bGaayjkaiaawMcaaiabgEna0oaabmaabaGaaGOmai abeE8aJnaaCaaaleqabaGaaGOmaaaakiabgUcaRiaaiwdaaiaawIca caGLPaaaaaa@48C8@ =3 x 2 y×2 x 2 +3 x 2 y×5+2y×2 x 2 +2y×5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG 4maiabeE8aJnaaCaaaleqabaGaaGOmaaaakiabeI8a5jabgEna0kaa ikdacqaHhpWydaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaIZaGaeq 4Xdm2aaWbaaSqabeaacaaIYaaaaOGaeqiYdKNaey41aqRaaGynaiab gUcaRiaaikdacqaHipqEcqGHxdaTcaaIYaGaeq4Xdm2aaWbaaSqabe aacaaIYaaaaOGaey4kaSIaaGOmaiabeI8a5jabgEna0kaaiwdaaaa@59C7@

=6 x 4 y+15 x 2 y+4 x 2 y+10y=6 x 4 y+19 x 2 y+10y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0JaaG OnaiabeE8aJnaaCaaaleqabaGaaGinaaaakiabeI8a5jabgUcaRiaa igdacaaI1aGaeq4Xdm2aaWbaaSqabeaacaaIYaaaaOGaeqiYdKNaey 4kaSIaaGinaiabeE8aJnaaCaaaleqabaGaaGOmaaaakiabeI8a5jab gUcaRiaaigdacaaIWaGaeqiYdKNaeyypa0JaaGOnaiabeE8aJnaaCa aaleqabaGaaGinaaaakiabeI8a5jabgUcaRiaaigdacaaI5aGaeq4X dm2aaWbaaSqabeaacaaIYaaaaOGaeqiYdKNaey4kaSIaaGymaiaaic dacqaHipqEaaa@5E85@

We conclude that in order to multiply a polynomial with another polynomial, we have to multiply each term of the first polynomial with every term of the second polynomial and then add the resulting products.

When multiplying a monomial with a polynomial or two polynomials with each other, we say that we are expanding these products and the result of the multiplication is called product expansion.

Trigonometric functions of angle 2a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiabeg 7aHbaa@3852@

The formulas expressing the trigonometric functions of this angle in terms of the trigonometric functions of angle a are specific expressions of the formulas in the former paragraph. Specifically, by replacing b with a, at the formulas of   sin( a+b ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacM gacaGGUbWaaeWaaeaacqaHXoqycqGHRaWkcqaHYoGyaiaawIcacaGL Paaaaaa@3E7A@ , cos( a+b ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbWaaeWaaeaacqaHXoqycqGHRaWkcqaHYoGyaiaawIcacaGL Paaaaaa@3E75@  and tan( a+b ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiDaiaacg gacaGGUbWaaeWaaeaacqaHXoqycqGHRaWkcqaHYoGyaiaawIcacaGL Paaaaaa@3E73@ we have

sin( 2a )=sin( a+a )=sin( a )×cos( a )+cos( a )×sin( a )=2×sin( a )×cos( a ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacM gacaGGUbWaaeWaaeaacaaIYaGaeqySdegacaGLOaGaayzkaaGaeyyp a0Jaci4CaiaacMgacaGGUbWaaeWaaeaacqaHXoqycqGHRaWkcqaHXo qyaiaawIcacaGLPaaacqGH9aqpciGGZbGaaiyAaiaac6gadaqadaqa aiabeg7aHbGaayjkaiaawMcaaiabgEna0kGacogacaGGVbGaai4Cam aabmaabaGaeqySdegacaGLOaGaayzkaaGaey4kaSIaci4yaiaac+ga caGGZbWaaeWaaeaacqaHXoqyaiaawIcacaGLPaaacqGHxdaTciGGZb GaaiyAaiaac6gadaqadaqaaiabeg7aHbGaayjkaiaawMcaaiabg2da 9iaaikdacqGHxdaTciGGZbGaaiyAaiaac6gadaqadaqaaiabeg7aHb GaayjkaiaawMcaaiabgEna0kGacogacaGGVbGaai4CamaabmaabaGa amyyaaGaayjkaiaawMcaaaaa@7578@ Consequently:  sin( 2a )=2×sin( a )×cos( a ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacM gacaGGUbWaaeWaaeaacaaIYaGaeqySdegacaGLOaGaayzkaaGaeyyp a0JaaGOmaiabgEna0kGacohacaGGPbGaaiOBamaabmaabaGaeqySde gacaGLOaGaayzkaaGaey41aqRaci4yaiaac+gacaGGZbWaaeWaaeaa cqaHXoqyaiaawIcacaGLPaaaaaa@4E9E@

cos( 2a )=cos( a+a )=cos( a )×cos( a )sin( a )×sin( a )= cos 2 a sin 2 a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbWaaeWaaeaacaaIYaGaeqySdegacaGLOaGaayzkaaGaeyyp a0Jaci4yaiaac+gacaGGZbWaaeWaaeaacqaHXoqycqGHRaWkcqaHXo qyaiaawIcacaGLPaaacqGH9aqpciGGJbGaai4Baiaacohadaqadaqa aiabeg7aHbGaayjkaiaawMcaaiabgEna0kGacogacaGGVbGaai4Cam aabmaabaGaeqySdegacaGLOaGaayzkaaGaeyOeI0Iaci4CaiaacMga caGGUbWaaeWaaeaacqaHXoqyaiaawIcacaGLPaaacqGHxdaTciGGZb GaaiyAaiaac6gadaqadaqaaiabeg7aHbGaayjkaiaawMcaaiabg2da 9iGacogacaGGVbGaai4CamaaCaaaleqabaGaaGOmaaaakiabeg7aHj abgkHiTiGacohacaGGPbGaaiOBamaaCaaaleqabaGaaGOmaaaakiab eg7aHbaa@7109@ = cos 2 a(1 cos 2 a)=2 cos 2 a1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaci 4yaiaac+gacaGGZbWaaWbaaSqabeaacaaIYaaaaOGaeqySdeMaeyOe I0IaaiikaiaaigdacqGHsislciGGJbGaai4BaiaacohadaahaaWcbe qaaiaaikdaaaGccqaHXoqycaGGPaGaeyypa0JaaGOmaiGacogacaGG VbGaai4CamaaCaaaleqabaGaaGOmaaaakiabeg7aHjabgkHiTiaaig daaaa@4E84@

=(1 sin 2 a) sin 2 a=12 sin 2 a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyypa0Jaai ikaiaaigdacqGHsislciGGZbGaaiyAaiaac6gadaahaaWcbeqaaiaa ikdaaaGccqaHXoqycaGGPaGaeyOeI0Iaci4CaiaacMgacaGGUbWaaW baaSqabeaacaaIYaaaaOGaeqySdeMaeyypa0JaaGymaiabgkHiTiaa ikdaciGGZbGaaiyAaiaac6gadaahaaWcbeqaaiaaikdaaaGccqaHXo qyaaa@4E93@

Additionally: cos( 2a )= cos 2 a sin 2 a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbWaaeWaaeaacaaIYaGaeqySdegacaGLOaGaayzkaaGaeyyp a0Jaci4yaiaac+gacaGGZbWaaWbaaSqabeaacaaIYaaaaOGaeqySde MaeyOeI0Iaci4CaiaacMgacaGGUbWaaWbaaSqabeaacaaIYaaaaOGa eqySdegaaa@4970@

Consequently:  cos( 2a )= cos 2 a sin 2 a=2 cos 2 a1=12 sin 2 a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbWaaeWaaeaacaaIYaGaeqySdegacaGLOaGaayzkaaGaeyyp a0Jaci4yaiaac+gacaGGZbWaaWbaaSqabeaacaaIYaaaaOGaeqySde MaeyOeI0Iaci4CaiaacMgacaGGUbWaaWbaaSqabeaacaaIYaaaaOGa eqySdeMaeyypa0JaaGOmaiGacogacaGGVbGaai4CamaaCaaaleqaba GaaGOmaaaakiabeg7aHjabgkHiTiaaigdacqGH9aqpcaaIXaGaeyOe I0IaaGOmaiGacohacaGGPbGaaiOBamaaCaaaleqabaGaaGOmaaaaki abeg7aHbaa@5B13@

tan(2a)= tana+tana 1tanatana = 2tana 1 tan 2 a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiDaiaacg gacaGGUbGaaiikaiaaikdacqaHXoqycaGGPaGaeyypa0ZaaSaaaeaa ciGG0bGaaiyyaiaac6gacqaHXoqycqGHRaWkciGG0bGaaiyyaiaac6 gacqaHXoqyaeaacaaIXaGaeyOeI0IaciiDaiaacggacaGGUbGaeqyS deMaciiDaiaacggacaGGUbGaeqySdegaaiabg2da9maalaaabaGaaG OmaiGacshacaGGHbGaaiOBaiabeg7aHbqaaiaaigdacqGHsislciGG 0bGaaiyyaiaac6gadaahaaWcbeqaaiaaikdaaaGccqaHXoqyaaaaaa@5F29@ , consequently tan(2a)= 2tana 1 tan 2 a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiDaiaacg gacaGGUbGaaiikaiaaikdacqaHXoqycaGGPaGaeyypa0ZaaSaaaeaa caaIYaGaciiDaiaacggacaGGUbGaeqySdegabaGaaGymaiabgkHiTi GacshacaGGHbGaaiOBamaaCaaaleqabaGaaGOmaaaakiabeg7aHbaa aaa@49C9@

cos(2a)=2 cos 2 a11+cos( 2a )=2 cos 2 a cos 2 a= 1+cos2a 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbGaaiikaiaaikdacqaHXoqycaGGPaGaeyypa0JaaGOmaiGa cogacaGGVbGaai4CamaaCaaaleqabaGaaGOmaaaakiabeg7aHjabgk HiTiaaigdacqGHuhY2caaIXaGaey4kaSIaci4yaiaac+gacaGGZbWa aeWaaeaacaaIYaGaeqySdegacaGLOaGaayzkaaGaeyypa0JaaGOmai GacogacaGGVbGaai4CamaaCaaaleqabaGaaGOmaaaakiabeg7aHjab gsDiBlGacogacaGGVbGaai4CamaaCaaaleqabaGaaGOmaaaakiabeg 7aHjabg2da9maalaaabaGaaGymaiabgUcaRiGacogacaGGVbGaai4C aiaaikdacqaHXoqyaeaacaaIYaaaaaaa@6782@

cos( 2a )=12 sin 2 a2 sin 2 a=1cos(2a)si n 2 a= 1cos( 2a ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbWaaeWaaeaacaaIYaGaeqySdegacaGLOaGaayzkaaGaeyyp a0JaaGymaiabgkHiTiaaikdaciGGZbGaaiyAaiaac6gadaahaaWcbe qaaiaaikdaaaGccqaHXoqycqGHuhY2caaIYaGaci4CaiaacMgacaGG UbWaaWbaaSqabeaacaaIYaaaaOGaeqySdeMaeyypa0JaaGymaiabgk HiTiGacogacaGGVbGaai4CaiaacIcacaaIYaGaeqySdeMaaiykaiab gsDiBlaacohacaGGPbGaaiOBamaaCaaaleqabaGaaGOmaaaakiabeg 7aHjabg2da9maalaaabaGaaGymaiabgkHiTiGacogacaGGVbGaai4C amaabmaabaGaaGOmaiabeg7aHbGaayjkaiaawMcaaaqaaiaaikdaaa aaaa@692E@

tan 2 a= sin 2 a cos 2 a = 1cos(2a) 2 1+cos( 2a ) 2 = 1cos(2a) 1+cos(2a) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiDaiaacg gacaGGUbWaaWbaaSqabeaacaaIYaaaaOGaeqySdeMaeyypa0ZaaSaa aeaaciGGZbGaaiyAaiaac6gadaahaaWcbeqaaiaaikdaaaGccqaHXo qyaeaaciGGJbGaai4BaiaacohadaahaaWcbeqaaiaaikdaaaGccqaH XoqyaaGaeyypa0ZaaSaaaeaadaWcaaqaaiaaigdacqGHsislciGGJb Gaai4BaiaacohacaGGOaGaaGOmaiabeg7aHjaacMcaaeaacaaIYaaa aaqaamaalaaabaGaaGymaiabgUcaRiGacogacaGGVbGaai4Camaabm aabaGaaGOmaiabeg7aHbGaayjkaiaawMcaaaqaaiaaikdaaaaaaiab g2da9maalaaabaGaaGymaiabgkHiTiGacogacaGGVbGaai4CaiaacI cacaaIYaGaeqySdeMaaiykaaqaaiaaigdacqGHRaWkciGGJbGaai4B aiaacohacaGGOaGaaGOmaiabeg7aHjaacMcaaaaaaa@6BD9@

If we know the trigonometric numbers of an angle, with the help of the formulas above, we are able to calculate the half angle trigonometric numbers. For example, the trigonometric numbers of angle 22.5°= 45° 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaaik dacaGGUaGaaGynaiabgclaWkabg2da9maalaaabaGaaGinaiaaiwda cqGHWcaSaeaacaaIYaaaaaaa@4007@  are calculated as follows:

sin 2 ( 22.5° )= 1cos( 45° ) 2 = 1 2 2 2 = 2 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4CaiaacM gacaGGUbWaaWbaaSqabeaacaaIYaaaaOWaaeWaaeaacaaIYaGaaGOm aiaac6cacaaI1aGaeyiSaalacaGLOaGaayzkaaGaeyypa0ZaaSaaae aacaaIXaGaeyOeI0Iaci4yaiaac+gacaGGZbWaaeWaaeaacaaI0aGa aGynaiabgclaWcGaayjkaiaawMcaaaqaaiaaikdaaaGaeyypa0ZaaS aaaeaacaaIXaGaeyOeI0YaaSaaaeaadaGcaaqaaiaaikdaaSqabaaa keaacaaIYaaaaaqaaiaaikdaaaGaeyypa0ZaaSaaaeaacaaIYaGaey OeI0YaaOaaaeaacaaIYaaaleqaaaGcbaGaaGOmaaaaaaa@54E2@ sin( 22.5° )= 2 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyO0H4Taci 4CaiaacMgacaGGUbWaaeWaaeaacaaIYaGaaGOmaiaac6cacaaI1aGa eyiSaalacaGLOaGaayzkaaGaeyypa0ZaaSaaaeaadaGcaaqaaiaaik dacqGHsisldaGcaaqaaiaaikdaaSqabaaabeaaaOqaaiaaikdaaaaa aa@45F6@

cos 2 (22.5°)= 1+cos( 45° ) 2 = 1+ 2 2 2 = 2+ 2 2 cos(22.5°)= 2+ 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGGZbWaaWbaaSqabeaacaaIYaaaaOGaaiikaiaaikdacaaIYaGa aiOlaiaaiwdacqGHWcaScaGGPaGaeyypa0ZaaSaaaeaacaaIXaGaey 4kaSIaci4yaiaac+gacaGGZbWaaeWaaeaacaaI0aGaaGynaiabgcla WcGaayjkaiaawMcaaaqaaiaaikdaaaGaeyypa0ZaaSaaaeaacaaIXa Gaey4kaSYaaSaaaeaadaGcaaqaaiaaikdaaSqabaaakeaacaaIYaaa aaqaaiaaikdaaaGaeyypa0ZaaSaaaeaacaaIYaGaey4kaSYaaOaaae aacaaIYaaaleqaaaGcbaGaaGOmaaaacqGHshI3ciGGJbGaai4Baiaa cohacaGGOaGaaGOmaiaaikdacaGGUaGaaGynaiabgclaWkaacMcacq GH9aqpdaWcaaqaamaakaaabaGaaGOmaiabgUcaRmaakaaabaGaaGOm aaWcbeaaaeqaaaGcbaGaaGOmaaaaaaa@644B@

Consequently: tan(22.5°)= 2 2 2+ 2 = 2 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciiDaiaacg gacaGGUbGaaiikaiaaikdacaaIYaGaaiOlaiaaiwdacqGHWcaScaGG PaGaeyypa0ZaaSaaaeaadaGcaaqaaiaaikdacqGHsisldaGcaaqaai aaikdaaSqabaaabeaaaOqaamaakaaabaGaaGOmaiabgUcaRmaakaaa baGaaGOmaaWcbeaaaeqaaaaakiabg2da9maakaaabaGaaGOmaaWcbe aakiabgkHiTiaaigdaaaa@48C4@ and cot(22.5°)= 2+ 2 2 2 = 2 +1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaci4yaiaac+ gacaGG0bGaaiikaiaaikdacaaIYaGaaiOlaiaaiwdacqGHWcaScaGG PaGaeyypa0ZaaSaaaeaadaGcaaqaaiaaikdacqGHRaWkdaGcaaqaai aaikdaaSqabaaabeaaaOqaamaakaaabaGaaGOmaiabgkHiTmaakaaa baGaaGOmaaWcbeaaaeqaaaaakiabg2da9maakaaabaGaaGOmaaWcbe aakiabgUcaRiaaigdaaaa@48BC@

Estimating unknown probability density functions and unknown parameters using the maximum likelihood estimation method

Let's suppose a problem of M classes with vectors of characteristics distributed following p(x| ω i ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiaacI cacqaHhpWydaabbaqaaiabeM8a3naaBaaaleaacqaH5oqAaeqaaOGa aiykaaGaay5bSdaaaa@3F42@ for i in 1 to Μ. Suppose that these likelihood functions are given in parametric form and that the respective parameters are forming vectors θ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUde3aaS baaSqaaiaadMgaaeqaaaaa@38C7@ that are unknown. In order to indicate the dependence of the probability density functions from these vectors, we write p(x| ω i ;θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiaacI cacqaHhpWydaabbaqaaiabeM8a3naaBaaaleaacqaH5oqAaeqaaOGa ai4oaiabeI7aXjaacMcaaiaawEa7aaaa@41B7@ .Our goal is to estimate the unknown parameters using a set of available vectors of characteristics out of each class.

Suppose that x 1 ,, x Ν MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Xdm2aaS baaSqaaiaaigdaaeqaaOGaaiilaiablAciljaacYcacqaHhpWydaWg aaWcbaGaeuyNd4eabeaaaaa@3E7D@   are random samples taken from the probability density function. We form the joint probability function p(X;θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiaacI cacqqHNoWqcaGG7aGaeqiUdeNaaiykaaaa@3C32@ . If we assume statistical independence between samples, we have

p(X;θ)=p( x 1 , x 2 ,, x N ;θ)= k=1 N p( x k ;θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaiaacI cacqqHNoWqcaGG7aGaeqiUdeNaaiykaiabg2da9iaacchacaGGOaGa eq4Xdm2aaSbaaSqaaiaaigdaaeqaaOGaaiilaiabeE8aJnaaBaaale aacaaIYaaabeaakiaacYcacqWIMaYscaGGSaGaeq4Xdm2aaSbaaSqa aiabf25aobqabaGccaGG7aGaeqiUdeNaaiykaiabg2da9maarahaba GaamiCaiaacIcacqaHhpWydaWgaaWcbaGaeqOUdSgabeaakiaacUda cqaH4oqCcaGGPaaaleaacqaH6oWAcqGH9aqpcaaIXaaabaGaeuyNd4 eaniabg+Givdaaaa@5E68@

This is a function of θ known as the likelihood function of θ in terms of X. The maximum likelihood estimation method estimates θ, so that the likelihood function is maximized at θ, meaning

θ ML =arg max θ k=1 N p( x k ;θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiUde3aaS baaSqaaiaad2eacaWGmbaabeaakiabg2da9iGacggacaGGYbGaai4z aiGac2gacaGGHbGaaiiEamaaBaaaleaacqaH4oqCaeqaaOWaaebCae aacaWGWbGaaiikaiabeE8aJnaaBaaaleaacqaH6oWAaeqaaOGaai4o aiabeI7aXjaacMcaaSqaaiabeQ7aRjabg2da9iaaigdaaeaacqqHDo Gta0Gaey4dIunaaaa@5193@

A necessary condition that must be satisfied by θ is to be a maximum point, so the derivative of the likelihood function in terms of θ, becomes 0, meaning:

k=1 N p( x k ;θ) θ =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHciITdaqeWbqaaiaadchacaGGOaGaeq4Xdm2aaSbaaSqaaiabeQ7a RbqabaGccaGG7aGaeqiUdeNaaiykaaWcbaGaeqOUdSMaeyypa0JaaG ymaaqaaiabf25aobqdcqGHpis1aaGcbaGaeyOaIyRaeqiUdehaaiab g2da9iaaicdaaaa@4BD2@

As a result of the increasing monotonicity of the of the logarithmic function, we define the logarithmic or likelihood function as:

L(θ)=ln k=1 N p( x k ;θ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaiaacI cacqaH4oqCcaGGPaGaeyypa0JaciiBaiaac6gadaqeWbqaaiaadcha caGGOaGaeq4Xdm2aaSbaaSqaaiabeQ7aRbqabaGccaGG7aGaeqiUde NaaiykaaWcbaGaeqOUdSMaeyypa0JaaGymaaqaaiabf25aobqdcqGH pis1aaaa@4C40@

Consequently, now we have the following:

L(θ) θ = k=1 N lnp( x k ;θ) θ = k=1 N 1 p( x k ;θ) × p( x k ;θ) θ =0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq GHciITcaWGmbGaaiikaiabeI7aXjaacMcaaeaacqGHciITcqaH4oqC aaGaeyypa0ZaaabCaeaadaWcaaqaaiabgkGi2kGacYgacaGGUbGaam iCaiaacIcacqaHhpWydaWgaaWcbaGaeqOUdSgabeaakiaacUdacqaH 4oqCcaGGPaaabaGaeyOaIyRaeqiUdehaaaWcbaGaeqOUdSMaeyypa0 JaaGymaaqaaiabf25aobqdcqGHris5aOGaeyypa0ZaaabCaeaadaWc aaqaaiaaigdaaeaacaWGWbGaaiikaiabeE8aJnaaBaaaleaacqaH6o WAaeqaaOGaai4oaiabeI7aXjaacMcaaaGaey41aq7aaSaaaeaacqGH ciITcaWGWbGaaiikaiabeE8aJnaaBaaaleaacqaH6oWAaeqaaOGaai 4oaiabeI7aXjaacMcaaeaacqGHciITcqaH4oqCaaaaleaacqaH6oWA cqGH9aqpcaaIXaaabaGaeuyNd4eaniabggHiLdGccqGH9aqpcaaIWa aaaa@77B5@