로그 탄젠트 적분(log tangent integral)

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

\(\int_0^{\pi}\frac{x\sin x}{1+\cos^2 x}dx=\frac{\pi^2}{4}\)

\(\int_0^{\pi}\frac{x\cos x}{1+\sin^2 x}dx=\ln^2(\sqrt{2}+1)-\frac{\pi^2}{4}\)

• 르장드르 카이 함수
  

 

 

메모

\(\int_{0}^{\infty}\frac{\ln(x^{2}+1)}{x^{2}+1}=\pi\ln2\)

http://cjackal.tistory.com/109

http://www.artofproblemsolving.com/Forum/viewtopic.php?t=340081

 

• 카탈란 상수
  

 

 

 

Gradshteyn and Ryzhik

• http://www.math.tulane.edu/~vhm/Table.html
  
• Directory of notes on the integrals in GR
  
• http://arxiv.org/find/math/1/au:+Moll_V/0/1/0/all/0/1
  
• The integrals in Gradshteyn and Ryzhik. Part 1: A family of logarithmic integrals.
  
  • Victor H. Moll
    

 

 

[[란덴변환(Landen's transformation)|]]

 

 

역사

• 수학사연표

 

 

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

• 대한수학회 수학 학술 용어집
  
  • http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
• 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

 

 

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)
• 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

• http://www.jstor.org/action/doBasicSearch?Query=

 

 

관련도서 및 추천도서

• Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals
  
  • George Boros and Victor Moll

• 도서내검색
  
  • http://books.google.com/books?q=
  • http://book.daum.net/search/contentSearch.do?query=
• 도서검색
  
  • http://books.google.com/books?q=
  • http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
  • http://book.daum.net/search/mainSearch.do?query=

 

 

관련기사

• 네이버 뉴스 검색 (키워드 수정)
  
  • http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
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  • http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=

 

 

블로그

• 수학과 잡담을 위한 소박한 장소
  
  • '오늘의 계산' 카테고리
  • sos440의 스프링노트 , 쓸만한 낙섲
• 오늘의 계산 12
  
  • 수학 잡담/오늘의 계산, 2008/08/10
• 구글 블로그 검색 http://blogsearch.google.com/blogsearch?q=
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