"로그 탄젠트 적분(log tangent integral)"의 두 판 사이의 차이
Pythagoras0 (토론 | 기여) |
Pythagoras0 (토론 | 기여) |
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194번째 줄: | 194번째 줄: | ||
* [http://www.amazon.com/Irresistible-Integrals-Symbolics-Experiments-Evaluation/dp/0521796369 Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals]<br> | * [http://www.amazon.com/Irresistible-Integrals-Symbolics-Experiments-Evaluation/dp/0521796369 Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals]<br> | ||
** George Boros and Victor Moll | ** George Boros and Victor Moll | ||
+ | [[분류:적분]] |
2013년 2월 11일 (월) 08:12 판
개요
- 로그 사인 적분 (log sine integrals)과 밀접하게 관련되어 있음
- 다음과 같은 정적분값의 계산\[\int_{\pi/4}^{\pi/2} \ln \tan x\, dx=G\], \(G\)는 카탈란 상수\[\int_{0}^{\pi/4} \ln \tan x\,dx=-G\]
\[\int_{\pi/4}^{\pi/2} \ln \ln \tan x\, dx=\frac{\pi}{2}\ln \left(\frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})}\sqrt{2\pi}\right)\]\[\int_{0}^{\pi} (\ln \tan \frac{x}{4})^2\,dx=\frac{\pi^3}{4}\]\[\int_0^{\infty}\frac{(\ln x)^2}{1+x^2} dx =\int_{0}^{\pi/2}(\ln \tan x)^2\,dx = \frac{ \pi^3}{8}\]
- '[Vardi1988] '참조
- 적분\[\int_{0}^{\infty}\frac{\ln(x^{2}+1)}{x^{2}+1}\,dx=\pi\ln2\] 에 대해서는 로그함수와 유리함수가 있는 정적분
증명
(보조정리)
\(\Gamma(s)\beta(s)=\int_{\pi/4}^{\pi/2} \ln^{s-1}\tan x\, dx\)
여기서 \(\Gamma(s)\)는 감마함수,\(\beta(s)\)는 디리클레 베타함수.
(증명)
\(F(s)=\sum_{n=1}^{\infty}\frac{f(n)}{n^s}\) 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.
\(f\)가 \(f(3)=-1\)인 주기가 4인 디리클레 캐릭터라면, \(q=4\), \(p(z)=z-z^3\)
따라서
\(\Gamma(s)\beta(s)=\int_0^{1}\frac{(\log\frac{1}{z})^{s-1}}{1+z^2} \,dz=\int_1^{\infty}\frac{(\log u)^{s-1}}{1+u^2} \,du=\int_{\pi/4}^{\pi/2} \ln^{s-1}\tan x\, dx\) ■
(따름정리1)
\(\int_{\pi/4}^{\pi/2} \ln \tan x\, dx=G\), G는 카탈란 상수.
(증명)
위에서 얻은 보조정리에 \(s=2\)를 적용하면,
\(\int_{\pi/4}^{\pi/2} \ln^{2-1}\tan x\, dx=\Gamma(2)\beta(2)=G\) ■
(따름정리2)
\(\int_{\pi/4}^{\pi/2} \ln \ln \tan x\, dx=\frac{\pi}{2}\ln \left(\frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})}\sqrt{2\pi}\right)\)
(증명)
\(\int_{\pi/4}^{\pi/2} \ln \ln \tan x\, dx=\frac{d}{ds}(\Gamma(s)\beta(s))|_{s=1}\)임을 보이자.
\(\frac{d}{ds}(\Gamma(s)\beta(s))=\frac{d}{ds}\int_1^{\infty}\frac{(\log u)^{s-1}}{1+u^2} \,du=\int_1^{\infty}\frac{(\log u)^{s-1}}{1+u^2}\log \log u \,du\)
\(s=1\) 일때,
\(\Gamma'(1)\beta(1)+\Gamma(1)\beta'(1)=\int_1^{\infty}\log \log u \,\frac{du}{1+u^2}=\int_{\pi/4}^{\pi/2} \ln \ln \tan x\, dx\)
이제 다이감마 함수(digamma function)와 디리클레 베타함수에서 얻은 결과를 사용하자.
\(\psi(x) =\frac{d}{dx} \ln{\Gamma(x)}= \frac{\Gamma'(x)}{\Gamma(x)}\), \(\psi(1) = -\gamma\,\!\). 따라서 \(\Gamma(1)=-\gamma\).
\(\beta'(1)=\frac{\pi}{4}\gamma+\frac{\pi}{2}\ln(\frac{\Gamma(3/4)}{\Gamma(1/4)}\sqrt{2\pi})\).
그러므로
\(\int_{\pi/4}^{\pi/2} \ln \ln \tan x\, dx=\Gamma'(1)\beta(1)+\Gamma(1)\beta'(1)= -\frac{\pi}{4}\gamma+\beta'(1)=\frac{\pi}{2}\ln \left(\frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})}\sqrt{2\pi}\right)\)
임이 증명된다. ■
역사
메모
\(\frac{24}{7\sqrt{7}}\int_{\pi/3}^{\pi/2}\ln|\frac{\tan t+\sqrt{7}}{\tan t-\sqrt{7}}|\,dt=L_{-7}(2)=1.15192547054449\cdots\)
개요
- 등차수열의 소수분포에 관한 디리클레 정리
- 디리클레 급수
- Hurwitz 제타함수
- 감마함수
- 다이로그 함수(dilogarithm )
- 란덴변환(Landen's transformation)
- 르장드르 카이 함수
- 모듈라 군, j-invariant and the singular moduli
- 카탈란 상수
수학용어번역
매스매티카 파일 및 계산 리소스
- https://docs.google.com/file/d/0B8XXo8Tve1cxNmRFU0Vyak14NGM/edit
- 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
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
- Alternative Evaluation of a ln tan Integral Arising in Quantum Field Theory
- Mark W. Coffey, Mathematical Physics, Analysis and Geometry, Volume 13, Number 2, 2010
- Mark W. Coffey, Mathematical Physics, Analysis and Geometry, Volume 13, Number 2, 2010
- [BBBZ2010]Experimental Mathematics and Mathematical Physics
- DH Bailey, JM Borwein, D Broadhurst, W Zudilin, 2010
- A class of logarithmic integrals
- Luis A. Medina 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)
- [Borwein and Boradhurst 1998]Determinations of rational Dedekind-zeta invariants of hyperbolic manifolds and Feynman knots and links
- J.M. Borwein, D.J. Broadhurst, 1998
- A class of logarithmic integrals
- Victor Adamchik, 1997
- [Vardi1988]Integrals, an Introduction to Analytic Number Theory
- Ilan Vardi, The American Mathematical Monthly, Vol. 95, No. 4 (Apr., 1988), pp. 308-315
관련도서
- Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals
- George Boros and Victor Moll