"로그 탄젠트 적분(log tangent integral)"의 두 판 사이의 차이
| 174번째 줄: | 174번째 줄: | ||
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| + | * [http://www.ams.org/notices/201004/rtx100400476p.pdf Seized Opportunities]<br> | ||
| + | ** Victor H. Moll, Notices of the AMS, Apr. 2010 | ||
* [http://www.springerlink.com/content/p2k0106727416271/?p=03915f5244d74523b6d36406299c80d5&pi=6 A class of logarithmic integrals]<br> | * [http://www.springerlink.com/content/p2k0106727416271/?p=03915f5244d74523b6d36406299c80d5&pi=6 A class of logarithmic integrals]<br> | ||
** Luis A. Medina1 and Victor H. Moll, The Ramanujan Journal, Volume 20, Number 1 / 2009년 10월 | ** Luis A. Medina1 and Victor H. Moll, The Ramanujan Journal, Volume 20, Number 1 / 2009년 10월 | ||
* [http://link.aip.org/link/?JMAPAQ/49/093508/1 Evaluation of a ln tan integral arising in quantum field theory]<br> | * [http://link.aip.org/link/?JMAPAQ/49/093508/1 Evaluation of a ln tan integral arising in quantum field theory]<br> | ||
| − | ** Mark W. Coffey, J. Math. Phys. 49, 093508 (2008) | + | ** Mark W. Coffey, J. Math. Phys. 49, 093508 (2008) |
* [http://arxiv.org/abs/0704.3872v2 The integrals in Gradshteyn and Ryzhik. Part 1: A family of logarithmic integrals.]<br> | * [http://arxiv.org/abs/0704.3872v2 The integrals in Gradshteyn and Ryzhik. Part 1: A family of logarithmic integrals.]<br> | ||
** Victor H. Moll<br> | ** Victor H. Moll<br> | ||
2010년 5월 21일 (금) 12:48 판
쇼1
\(\int_{\pi/4}^{\pi/2} \ln \ln \tan x\, dx=\frac{\pi}{2}\ln{\frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})}\sqrt{2\pi}\)
증명
[Vardi1988] 참조
\(\int_{\pi/4}^{\pi/2} \ln \ln \tan x\, dx=\frac{d}{ds}\Gamma(s)\beta(s)|_{s=1}\)임을 보이자. 여기서 \(\Gamma(s)\)는 감마함수,\(\beta(s)\)는 디리클레 베타함수.
이것이 참이라면, Digamma 함수와 디리클레 베타함수에서 얻은 결과를 사용하여 계산할 수 있다.
\(\int_{\pi/4}^{\pi/2} \ln \ln \tan x\, dx=\Gamma'(1)\beta(1)+\Gamma(1)\beta'(1)= -\frac{\pi}{4}\gamma+\beta'(1)\)
\(\psi(x) =\frac{d}{dx} \ln{\Gamma(x)}= \frac{\Gamma'(x)}{\Gamma(x)}\), \(\psi(1) = -\gamma\,\!\)
\(\beta'(1)=\frac{\pi}{4}\gamma+\frac{\pi}{2}\ln(\frac{\Gamma(3/4)}{\Gamma(1/4)}\sqrt{2\pi})\)
\(F(s)=\sum_{n=1}^{\infty}\frac{f(n)}{n^s}\) 라 하자.
\(\Gamma(s)F(s)=\int_0^{\infty}(\sum_{n=1}^{\infty}f(n)e^{-nt})t^{s-1}\,dt\)
\(z=e^{-t}\) 로 치환하면,
\(\Gamma(s)F(s)=\int_0^{1}(\sum_{n=1}^{\infty}f(n)z^n)(\log\frac{1}{z})^{s-1}\,\frac{dz}{z}\)
만약 \(f(n+q)=f(n)\) 을 만족하면 (가령 디리클레 캐릭터의 경우)
\(p(z)=\sum_{n=1}^{q-1}f(n)z^n\)라면, \(\sum_{n=1}^{\infty}f(n)z^n=\frac{p(z)}{1-z^q}\) 로 쓸 수 있다.
이를 이용하면,
\(\Gamma(s)F(s)=\int_0^{1}\frac{p(z)(\log\frac{1}{z})^{s-1}}{1-z^q}\,\frac{dz}{z}\) 를 얻는다.
\(\frac{d}{ds}\Gamma(s)F(s)=\int_0^{1}\frac{p(z)(\log\frac{1}{z})^{s-1}}{1-z^q}\log \log\frac{1}{z} \,\frac{dz}{z}\)
\(s=1\) 에서 \(F(s)\)가 미분가능하다면,
\(\Gamma'(1)F(1)+\Gamma(1)F'(1)=\int_0^{1}\frac{p(z)}{1-z^q}\log \log\frac{1}{z} \,\frac{dz}{z}\)
\(f\)가 \(f(3)=-1\)인 주기가 4인 디리클레 캐릭터라면, \(q=4\), \(p(z)=z-z^3\)
따라서
\(\int_0^{1}\frac{z-z^3}{1-z^4}\log \log\frac{1}{z} \,\frac{dz}{z}=\int_0^{1}\log \log\frac{1}{z} \,\frac{dz}{1+z^2}=\int_1^{\infty}\log \log u \,\frac{du}{1+u^2}\)
\(=\int_{\pi/4}^{\pi/2} \ln \ln \tan x\, dx\)
그러므로
\(\int_{\pi/4}^{\pi/2} \ln \ln \tan x\, dx=\beta'(1)- \frac{\pi}{4}\gamma=\frac{\pi}{2}\ln{\frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})}\sqrt{2\pi}\)
임이 증명된다.
(증명끝)
쇼2
\(\int_0^{\pi}\frac{x\sin x}{1+\cos^2 x}dx=\frac{\pi^2}{4}\)
\(\int_0^{\pi}\frac{x\cos x}{1+\sin^2 x}dx=\ln^2(\sqrt{2}+1)-\frac{\pi^2}{4}\)
메모
\(\int_{0}^{\infty}\frac{\ln(x^{2}+1)}{x^{2}+1}=\pi\ln2\)
http://cjackal.tistory.com/109
Gradshteyn and Ryzhik
[[란덴변환(Landen's transformation)|]]
재미있는 사실
역사
메모
- http://www.wolframalpha.com/input/?i=integrate_0^(pi)+x+cos+x+%2F(1%2Bsin^2+x)
- http://www.wolframalpha.com/input/?i=log^2+(1%2Bsqrt(2))+-pi^2%2F4
- http://www.wolframalpha.com/input/?i=integrate+(tan+x%2B1)/sqrt(tan^2+x+%2B1)dx
- http://www.wolframalpha.com/input/?i=integrate+(tan+x-1)/sqrt(tan^2+x+%2B1)dx
-
관련된 다른 주제들
- 등차수열의 소수분포에 관한 디리클레 정리
- 디리클레 급수
- Hurwitz 제타함수
- 감마함수
- Dilogarithm
- 란덴변환(Landen's transformation)
- 르장드르 카이 함수
- 모듈라 군, j-invariant and the singular moduli
수학용어번역
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
관련논문
- Seized Opportunities
- Victor H. Moll, Notices of the AMS, Apr. 2010
- A class of logarithmic integrals
- Luis A. Medina1 and Victor H. Moll, The Ramanujan Journal, Volume 20, Number 1 / 2009년 10월
- Evaluation of a ln tan integral arising in quantum field theory
- Mark W. Coffey, J. Math. Phys. 49, 093508 (2008)
- The integrals in Gradshteyn and Ryzhik. Part 1: A family of logarithmic integrals.
- Victor H. Moll
- Victor H. Moll
- On Some Integrals Involving the Hurwitz Zeta Function: Part 1
- Olivier Espinosa and Victor H. Moll
- On Some Integrals Involving the Hurwitz Zeta Function: Part 2
- Olivier Espinosa and Victor H. Moll
- A class of logarithmic integrals
- Victor Adamchik, 1997
- 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
- [Vardi1988]Integrals, an Introduction to Analytic Number Theory
- Ilan Vardi, The American Mathematical Monthly, Vol. 95, No. 4 (Apr., 1988), pp. 308-315
관련도서 및 추천도서
- Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals
- George Boros and Victor Moll
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- 도서검색
관련기사
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