로그 사인 적분 (log sine integrals)

이 항목의 스프링노트 원문주소

• 로그 사인 적분 (log sine integrals)
  

 

 

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

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

 

 

 

special values

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

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

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

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

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

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

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

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

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

 

 

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
  
•  
  

 

 

 

관련된 항목들

• 다이로그 함수(dilogarithm )
  
• 로바체프스키와 클라우센 함수
  
• 정수에서의 리만제타함수의 값
  
• 중심이항계수(central binomial coefficient)
  

 

 

수학용어번역

• 단어사전 http://www.google.com/dictionary?langpair=en%7Cko&q=
• 발음사전 http://www.forvo.com/search/
• 대한수학회 수학 학술 용어집
  
  • http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
• 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
  
  • http://www.research.att.com/~njas/sequences/?q=

 

 

관련논문

• 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 Evaluation of log-sine Integrals
  
  • L. Lewin The Mathematical Gazette, Vol. 42, No. 340 (May, 1958), pp. 125-128
    
• http://www.jstor.org/stable/3609410
• http://www.jstor.org/action/doBasicSearch?Query=
• http://www.ams.org/mathscinet
• http://dx.doi.org/

 

 

관련도서

• 도서내검색
  
  • http://books.google.com/books?q=
  • http://book.daum.net/search/contentSearch.do?query=
• 도서검색
  
  • http://books.google.com/books?q=
  • http://book.daum.net/search/mainSearch.do?query=
  • 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=
  • http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
  • http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=

 

 

블로그

• 구글 블로그 검색
  
  • http://blogsearch.google.com/blogsearch?q=
• 네이버 오늘의과학
• 수학동아
• Mathematical Moments from the AMS
• BetterExplained