"로그 사인 적분 (log sine integrals)"의 두 판 사이의 차이

수학노트
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<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">개요</h5>
 
<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">개요</h5>
  
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<math>\operatorname{Ls}_{a+b,a}(\theta)=-\int_{0}^{\theta}x^a\log^{b-1}}|2\sin \frac{x}{2}|\,dx</math>
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*  정의<br><math>\operatorname{Ls}_{a+b,a}(\theta)=-\int_{0}^{\theta}x^a\log^{b-1}}|2\sin \frac{x}{2}|\,dx</math><br>
  
 
* [[로바체프스키 함수|클라우센 함수]]의 일반화로 볼 수 있다<br><math>\operatorname{Cl}_2(\theta)=-\int_0^{\theta} \ln |2\sin \frac{t}{2}| \,dt=\sum_{n=1}^{\infty}\frac{\sin (n\theta)}{n^2}</math><br>
 
* [[로바체프스키 함수|클라우센 함수]]의 일반화로 볼 수 있다<br><math>\operatorname{Cl}_2(\theta)=-\int_0^{\theta} \ln |2\sin \frac{t}{2}| \,dt=\sum_{n=1}^{\infty}\frac{\sin (n\theta)}{n^2}</math><br>
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<math>\int_{0}^{\pi/2}\log(\sin x)\,dx=-\frac{\pi\log 2}{2}</math>
 
<math>\int_{0}^{\pi/2}\log(\sin x)\,dx=-\frac{\pi\log 2}{2}</math>
  
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<math>\int_{0}^{\pi/2}\log(\sin x)\,dx=-\frac{\pi\log 2}{2}</math>
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<math>\int_{0}^{\pi/2}\log^2(\sin x)\,dx=\frac{\pi}{2}(\log 2)^2+\frac{\pi^3}{24}</math>
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<math>\operatorname{Ls}_3(\pi)=-\int_{0}^{\pi}\log^2(2\sin \frac{x}{2})\,dx=-\frac{\pi^3}{12}</math>
 
  
 
<math>\operatorname{Ls}_2(\pi)=-\int_{0}^{\pi}\log(2\sin \frac{x}{2})\,dx=0</math>
 
<math>\operatorname{Ls}_2(\pi)=-\int_{0}^{\pi}\log(2\sin \frac{x}{2})\,dx=0</math>
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* [[다이로그 함수(dilogarithm)|다이로그 함수(dilogarithm )]]<br>
 
* [[다이로그 함수(dilogarithm)|다이로그 함수(dilogarithm )]]<br>
 
* [[로바체프스키 함수|로바체프스키와 클라우센 함수]]<br>
 
* [[로바체프스키 함수|로바체프스키와 클라우센 함수]]<br>
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* [[정수에서의 리만제타함수의 값]]<br>
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* [[중심이항계수(central binomial coefficient)]]<br>
  
 
 
 
 
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<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">관련논문</h5>
 
<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">관련논문</h5>
  
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*  Some wonderful formulas ... an introduction to polylogarithms<br>
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** A.J. Van der Poorten, Queen's papers in Pure and Applied Mathematics, 54 (1979), 269-286
 
* [http://www.jstor.org/stable/3609410 On the Evaluation of log-sine Integrals]<br>
 
* [http://www.jstor.org/stable/3609410 On the Evaluation of log-sine Integrals]<br>
 
**  L. Lewin The Mathematical Gazette, Vol. 42, No. 340 (May, 1958), pp. 125-128<br>
 
**  L. Lewin The Mathematical Gazette, Vol. 42, No. 340 (May, 1958), pp. 125-128<br>
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*  Some wonderful formulas ... an introduction to polylogarithms<br>
 
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** A.J. Van der Poorten, Queen's papers in Pure and Applied Mathematics, 54 (1979), 269-286
 
 
* http://www.jstor.org/stable/3609410
 
* http://www.jstor.org/stable/3609410
 
* http://www.jstor.org/action/doBasicSearch?Query=
 
* http://www.jstor.org/action/doBasicSearch?Query=

2010년 6월 13일 (일) 16:06 판

이 항목의 스프링노트 원문주소

 

 

개요
  • 정의
    \(\operatorname{Ls}_{a+b,a}(\theta)=-\int_{0}^{\theta}x^a\log^{b-1}}|2\sin \frac{x}{2}|\,dx\)
  • 클라우센 함수의 일반화로 볼 수 있다
    \(\operatorname{Cl}_2(\theta)=-\int_0^{\theta} \ln |2\sin \frac{t}{2}| \,dt=\sum_{n=1}^{\infty}\frac{\sin (n\theta)}{n^2}\)

 

 

\(\int_{0}^{1-e^{i\theta}}\log^{n-1}z\frac{dz}{1-z}=-i\int_{0}^{\theta}(\frac{i}{2}(x-\pi)+\log|2\sin \frac{x}{2}|)^{n-1}\,dx \)\(=-\int_{0}^{\theta}x^a\log^{b-1}}|2\sin \frac{x}{2}|\,dx\)

 

 

special values의 생성함수
  • 정의
    \(\operatorname{Ls}_{n}(\pi)=-\int_{0}^{\pi}\log^{n-1}}(2\sin \frac{x}{2})\,dx\)
  • 생성함수
    \(I(x)=\int_{0}^{\pi}e^{x\log(2\sin \frac{1}{2}\theta)}d\theta =\sum_{n=0}^{\infty}\int_{0}^{\pi}\frac{x^n}{n!}\log^n(2\sin\frac{1}{2}\theta)d\theta=-\sum_{n=0}^{\infty}\frac{x^n}{n!}\operatorname{Ls}_{n+1}(\pi)\)
    \(I(x)=\frac{\pi\Gamma(1+x)}{(\Gamma(1+\frac{1}{2}x))^2}\)

 

점화식

\(\operatorname{Ls}_{m+2}(\pi)=(-1)^{m}m[\pi(1-2^{-m})\zeta(m+1)-(1-2^{2-m})\zeta(m-1)\operatorname{Ls}_{3}(\pi)/2!+(1-2^{3-m})\zeta(m-2)\operatorname{Ls}_{4}(\pi)/3!+(-1)^{m}\cdots+(1-1/2})\zeta(2)\operatorname{Ls}_{m}(\pi)/(m-1)!]\)

 

 

 

special values

\(\int_{0}^{\pi/2}\log(\sin x)\,dx=-\frac{\pi\log 2}{2}\)

\(\int_{0}^{\pi/2}\log^2(\sin x)\,dx=\frac{\pi}{2}(\log 2)^2+\frac{\pi^3}{24}\)

\(\operatorname{Ls}_2(\pi)=-\int_{0}^{\pi}\log(2\sin \frac{x}{2})\,dx=0\)

\(\operatorname{Ls}_3(\pi)=-\int_{0}^{\pi}\log^2(2\sin \frac{x}{2})\,dx=-\frac{\pi^3}{12}\)

\(\operatorname{Ls}_4(\pi)=-\int_{0}^{\pi}\log^3(2\sin \frac{x}{2})\,dx=\frac{3\pi}{2}\zeta(3)\)

\(\operatorname{Ls}_5(\pi)=-\int_{0}^{\pi}\log^4(2\sin \frac{x}{2})\,dx=-\frac{19\pi^5}{240}\)

\(\operatorname{Ls}_6(\pi)=-\int_{0}^{\pi}\log^5(2\sin \frac{x}{2})\,dx=\frac{45\pi}{2}\zeta(5)+\frac{5\pi^3}{4}\zeta(3)\)\(\operatorname{Ls}_7(\pi)=-\int_{0}^{\pi}\log^6(2\sin \frac{x}{2})\,dx=-\frac{45\pi}{2}\zeta^2(3)-\frac{275\pi^7}{1344}\)

\(\int_{0}^{\pi/3}\log^2(2\sin \frac{x}{2})\,dx=\frac{7\pi^3}{108}\)

\(\int_{0}^{\pi/3}x\log^2(2\sin \frac{x}{2})\,dx=\frac{17\pi^4}{6480}\)

 

 

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