"로그 사인 적분 (log sine integrals)"의 두 판 사이의 차이
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<h5 style="LINE-HEIGHT: 2em; MARGIN: 0px;">로그사인 정적분</h5> | <h5 style="LINE-HEIGHT: 2em; MARGIN: 0px;">로그사인 정적분</h5> | ||
| − | * | + | * 다음 정적분의 값<br><math>\operatorname{Ls}_{n}(\pi)=-\int_{0}^{\pi}\log^{n-1}}(2\sin \frac{x}{2})\,dx</math><br> |
* 생성함수<br><math>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)</math><br> | * 생성함수<br><math>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)</math><br> | ||
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여기서 [[감마함수]]의 곱셈공식 <math>2^{2z}\Gamma(z) \; \Gamma\left(z + \frac{1}{2}\right) = 2\sqrt{\pi}\;\Gamma(2z)</math> 을 이용하면, 우변을 정리하여 원하는 식을 얻는다. ■ | 여기서 [[감마함수]]의 곱셈공식 <math>2^{2z}\Gamma(z) \; \Gamma\left(z + \frac{1}{2}\right) = 2\sqrt{\pi}\;\Gamma(2z)</math> 을 이용하면, 우변을 정리하여 원하는 식을 얻는다. ■ | ||
| − | + | * 점화식<br><math>\operatorname{Ls}_{m+2}(\pi)=(-1)^{m}m![\pi(1-2^{-m})\zeta(m+1)-\sum_{k=2}^{m-1}(-1)^{k}\frac{1-2^{k-m}}{k!}\zeta(m-k+1)\operatorname{Ls}_{k+1}(\pi)</math><br> | |
| − | + | * 이 정적분은 <math>\ln 2</math>와 <math>\zeta(n), n\geq 2</math> 의 다항식으로 표현할 수 있다'''[Bowman1947]'''<br> <br> | |
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| − | <math>\operatorname{Ls}_{m+2}(\pi)=(-1)^{m}m![\pi(1-2^{-m})\zeta(m+1)-\sum_{k=2}^{m-1}(-1)^{k}\frac{1-2^{k-m}}{k!}\zeta(m-k+1)\operatorname{Ls}_{k+1}(\pi)</math> | ||
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* [[정수에서의 리만제타함수의 값]]<br> | * [[정수에서의 리만제타함수의 값]]<br> | ||
* [[폴리로그 함수(polylogarithm)]]<br> | * [[폴리로그 함수(polylogarithm)]]<br> | ||
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* [[중심이항계수(central binomial coefficient)]]<br> | * [[중심이항계수(central binomial coefficient)]]<br> | ||
* [[오일러 베타적분(베타함수)|오일러 베타적분]]<br> | * [[오일러 베타적분(베타함수)|오일러 베타적분]]<br> | ||
| 160번째 줄: | 152번째 줄: | ||
<h5 style="LINE-HEIGHT: 3.42em; MARGIN: 0px; FONT-FAMILY: 'malgun gothic', dotum, gulim, sans-serif; BACKGROUND-POSITION: 0px 100%; COLOR: rgb(34,61,103); FONT-SIZE: 1.16em;">관련논문</h5> | <h5 style="LINE-HEIGHT: 3.42em; MARGIN: 0px; FONT-FAMILY: 'malgun gothic', dotum, gulim, sans-serif; BACKGROUND-POSITION: 0px 100%; COLOR: rgb(34,61,103); FONT-SIZE: 1.16em;">관련논문</h5> | ||
| − | * [http://dx.doi.org/10.1016/S0377-0427(03)00438-2 On some log-cosine integrals related to ζ(3), ζ(4), and ζ(6)]<br> | + | * [http://dx.doi.org/10.1016/S0377-0427(03)00438-2 On some log-cosine integrals related to ζ(3), ζ(4), and ζ(6)]<br>Mark W. Coffey, 2003<br> |
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| + | * [http://www.jstor.org/stable/2160718 On an Intriguing Integral and Some Series Related to ζ(4)]<br> | ||
| + | ** David Borwein and Jonathan M. Borwein, Proceedings of the American Mathematical Society, Vol. 123, No. 4 (Apr., 1995), pp. 1191-1198 | ||
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* Some wonderful formulas ... an introduction to polylogarithms<br> | * Some wonderful formulas ... an introduction to polylogarithms<br> | ||
** A.J. Van der Poorten, Queen's papers in Pure and Applied Mathematics, 54 (1979), 269-286 | ** A.J. Van der Poorten, Queen's papers in Pure and Applied Mathematics, 54 (1979), 269-286 | ||
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* [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> | ||
| − | * [http://dx.doi.org/10.1112/jlms/s1-22.3.172 Note on the Integral] <br> | + | * '''[Bowman1947]'''[http://dx.doi.org/10.1112/jlms/s1-22.3.172 Note on the Integral] <br> |
** F. Bowman, J. London Math. Soc. 1947 s1-22: 172-173<br> | ** F. Bowman, J. London Math. Soc. 1947 s1-22: 172-173<br> | ||
* http://www.jstor.org/stable/3609410 | * http://www.jstor.org/stable/3609410 | ||
2010년 6월 16일 (수) 19:14 판
이 항목의 스프링노트 원문주소
개요
- 정의
\(\operatorname{Ls}_{a+b,a}(\theta)=-\int_{0}^{\theta}x^a\log^{b-1}}|2\sin \frac{x}{2}|\,dx\)
\(\operatorname{Ls}_{n}(\theta)=-\int_{0}^{\theta}\log^{n-1}}(2\sin \frac{x}{2})\,dx\)
- 클라우센 함수의 일반화로 볼 수 있다
\(\operatorname{Cl}_2(\theta)=-\int_0^{\theta} \ln |2\sin \frac{t}{2}| \,dt=\sum_{n=1}^{\infty}\frac{\sin (n\theta)}{n^2}\)
\(\int_{0}^{1-e^{i\theta}}\log^{n-1}z\frac{dz}{1-z}=-i\int_{0}^{\theta}(\frac{i}{2}(x-\pi)+\log|2\sin \frac{x}{2}|)^{n-1}\,dx \)\(=-\int_{0}^{\theta}x^a\log^{b-1}}|2\sin \frac{x}{2}|\,dx\)
로그사인 정적분
- 다음 정적분의 값
\(\operatorname{Ls}_{n}(\pi)=-\int_{0}^{\pi}\log^{n-1}}(2\sin \frac{x}{2})\,dx\) - 생성함수
\(I(x)=\int_{0}^{\pi}e^{x\log(2\sin \frac{1}{2}\theta)}d\theta =\sum_{n=0}^{\infty}\int_{0}^{\pi}\frac{x^n}{n!}\log^n(2\sin\frac{1}{2}\theta)d\theta=-\sum_{n=0}^{\infty}\frac{x^n}{n!}\operatorname{Ls}_{n+1}(\pi)\)
(정리)
\(I(x)=\frac{\pi\Gamma(1+x)}{(\Gamma(1+\frac{1}{2}x))^2}\)
(증명)
오일러 베타적분 의 결과를 이용하자.
\(\int_0^{\frac{\pi}{2}}\sin^{p}\theta{d\theta}= \frac{1}{2}B(\frac{p+1}{2},\frac{1}{2})=\frac{\sqrt{\pi}\Gamma(\frac{p}{2}+\frac{1}{2})}{2\Gamma(\frac{p}{2}+1)}\)
\(I(x)=\int_{0}^{\pi}e^{x\log(2\sin \frac{1}{2}\theta)}d\theta =\int_{0}^{\pi}(2\sin \frac{1}{2}\theta)^{x}\,d\theta=2^{x+1}\int_{0}^{\pi/2}\sin^{x}t\,dt=\sqrt{\pi}\frac{2^x\Gamma(\frac{x}{2}+\frac{1}{2})}{\Gamma(\frac{x}{2}+1)}\)
여기서 감마함수의 곱셈공식 \(2^{2z}\Gamma(z) \; \Gamma\left(z + \frac{1}{2}\right) = 2\sqrt{\pi}\;\Gamma(2z)\) 을 이용하면, 우변을 정리하여 원하는 식을 얻는다. ■
- 점화식
\(\operatorname{Ls}_{m+2}(\pi)=(-1)^{m}m![\pi(1-2^{-m})\zeta(m+1)-\sum_{k=2}^{m-1}(-1)^{k}\frac{1-2^{k-m}}{k!}\zeta(m-k+1)\operatorname{Ls}_{k+1}(\pi)\) - 이 정적분은 \(\ln 2\)와 \(\zeta(n), n\geq 2\) 의 다항식으로 표현할 수 있다[Bowman1947]
special values
\(\int_{0}^{\pi/2}\log(\sin x)\,dx=-\frac{\pi\log 2}{2}\)
\(\int_{0}^{\pi/2}\log^2(\sin x)\,dx=\frac{\pi}{2}(\log 2)^2+\frac{\pi^3}{24}\)
\(\operatorname{Ls}_2(\pi)=-\int_{0}^{\pi}\log(2\sin \frac{x}{2})\,dx=0\)
\(\operatorname{Ls}_3(\pi)=-\int_{0}^{\pi}\log^2(2\sin \frac{x}{2})\,dx=-\frac{\pi^3}{12}\)
\(\operatorname{Ls}_4(\pi)=-\int_{0}^{\pi}\log^3(2\sin \frac{x}{2})\,dx=\frac{3\pi}{2}\zeta(3)\)
\(\operatorname{Ls}_5(\pi)=-\int_{0}^{\pi}\log^4(2\sin \frac{x}{2})\,dx=-\frac{19\pi^5}{240}\)
\(\operatorname{Ls}_6(\pi)=-\int_{0}^{\pi}\log^5(2\sin \frac{x}{2})\,dx=\frac{45\pi}{2}\zeta(5)+\frac{5\pi^3}{4}\zeta(3)\)\(\operatorname{Ls}_7(\pi)=-\int_{0}^{\pi}\log^6(2\sin \frac{x}{2})\,dx=-\frac{45\pi}{2}\zeta^2(3)-\frac{275\pi^7}{1344}\)
\(\int_{0}^{\pi/3}\log^2(2\sin \frac{x}{2})\,dx=\frac{7\pi^3}{108}\)
\(\int_{0}^{\pi/3}x\log^2(2\sin \frac{x}{2})\,dx=\frac{17\pi^4}{6480}\)
재미있는 사실
- Math Overflow http://mathoverflow.net/search?q=
- 네이버 지식인 http://kin.search.naver.com/search.naver?where=kin_qna&query=
역사
- http://www.google.com/search?hl=en&tbs=tl:1&q=log+sine+integral
- http://www.google.com/search?hl=en&tbs=tl:1&q=
- 수학사연표
메모
- http://www.wolframalpha.com/input/?i=integrate+(log+(2sin(x/2)))^2+dx+from+0+to+pi/3
- http://arxiv.org/abs/hep-ph/0411100v2
- http://mathworld.wolfram.com/RamanujanLog-TrigonometricIntegrals.html
관련된 항목들
- 다이로그 함수(dilogarithm )
- 로바체프스키와 클라우센 함수
- 정수에서의 리만제타함수의 값
- 폴리로그 함수(polylogarithm)
- 중심이항계수(central binomial coefficient)
- 오일러 베타적분
수학용어번역
- 단어사전 http://www.google.com/dictionary?langpair=en%7Cko&q=
- 발음사전 http://www.forvo.com/search/
- 대한수학회 수학 학술 용어집
- 남·북한수학용어비교
- 대한수학회 수학용어한글화 게시판
사전 형태의 자료
- http://ko.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/
- http://www.wolframalpha.com/input/?i=
- NIST Digital Library of Mathematical Functions
- The On-Line Encyclopedia of Integer Sequences
관련논문
- On some log-cosine integrals related to ζ(3), ζ(4), and ζ(6)
Mark W. Coffey, 2003
- On an Intriguing Integral and Some Series Related to ζ(4)
- David Borwein and Jonathan M. Borwein, Proceedings of the American Mathematical Society, Vol. 123, No. 4 (Apr., 1995), pp. 1191-1198
- Some wonderful formulas ... an introduction to polylogarithms
- A.J. Van der Poorten, Queen's papers in Pure and Applied Mathematics, 54 (1979), 269-286
- On the value of a logarithmic-trigonometric integral
- K. S. Kölbig, 1970
- On the Evaluation of log-sine Integrals
- L. Lewin The Mathematical Gazette, Vol. 42, No. 340 (May, 1958), pp. 125-128
- L. Lewin The Mathematical Gazette, Vol. 42, No. 340 (May, 1958), pp. 125-128
- [Bowman1947]Note on the Integral
- F. Bowman, J. London Math. Soc. 1947 s1-22: 172-173
- F. Bowman, J. London Math. Soc. 1947 s1-22: 172-173
- http://www.jstor.org/stable/3609410
- http://www.jstor.org/action/doBasicSearch?Query=
- http://www.ams.org/mathscinet
- http://dx.doi.org/10.1007/BF01935325
관련도서
- 도서내검색
- 도서검색
관련기사
- 네이버 뉴스 검색 (키워드 수정)