클라우센 함수(Clausen function)

 

 

개요

 

•  
  클라우센 함수
  \(\operatorname{Cl}_2(\theta)=-\int_0^{\theta} \ln |2\sin \frac{t}{2}| \,dt=\sum_{n=1}^{\infty}\frac{\sin (n\theta)}{n^2}\)
  로그 사인 적분 (log sine integrals) 으로 일반화된다
   
  
• 로바체프스키 함수 와의 관계
  \(Cl_2(2\theta)=2\Lambda(\theta)\)
  

 

 

 

 

dilogarithm 함수와의 관계

• dilogarithm 함수는 복소수 \(|z|<1\)에 대하여 다음과 같이 정의됨
• \(\operatorname{Li}_2(z)= \sum_{n=1}^\infty {z^n \over n^2}\)
  \(|z|\leq 1\) 에서 고르게 수렴하는 급수이므로, \(|z|\leq 1\)에서 연속
  
• \(z=e^{2i\theta}\), \(0 \leq \theta \leq \pi\) 일 때,
  \(\operatorname{Li}_2(e^{2i\theta})= \sum_{n=1}^\infty \frac{e^{2in\theta}}{n^2}=\sum_{n=1}^\infty \frac{\cos 2n\theta}{n^2}+i\sum_{n=1}^\infty \frac{\sin 2n\theta}{n^2}\)
  

\(0 \leq \theta \leq 2\pi\) 일때, \(\mathfrak{I}(\operatorname{Li}_2(e^{i\theta}))=\sum_{n=1}^\infty \frac{\sin n\theta}{n^2}=Cl_2(\theta)\)

 

 

 

special values

• \(\operatorname{Cl}_2(\frac{\pi}{2})=G\)
  \(G\)는 카탈란 상수
  
• http://mathworld.wolfram.com/GiesekingsConstant.html
  

 

 

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=
• 수학사연표
•  

 

 

메모

\(\int_{0}^{\pi/3}\operatorname{Cl}_2(x)\,dx=\frac{2}{3}\zeta(3)\)

 

 

 

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/Clausen_function
• http://mathworld.wolfram.com/ClausensIntegral.html
• 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

• A dilogarithmic integral arising in quantum field theory
  
  • Djurdje Cvijović, J. Math. Phys. 50, 023515 (2009)
• On a three-dimensional symmetric Ising tetrahedron and contributions to the theory of the dilogarithm and Clausen functions
  
  • Mark W. Coffey, J. Math. Phys. 49, 043510 (2008); doi:10.1063/1.2902996
• Evaluation of a ln tan integral arising in quantum field theory
  
  • Mark W. Coffey, J. Math. Phys. 49, 093508 (2008); doi:10.1063/1.2981311
• [1]Formulae concerning the computation of the Clausen integral Cl2(θ)
  
  • C.C. Grosjean, J. Comput. Appl. Math. 11 (1984), pp. 331–342
• On the Clausen integral Cl2(Θ) and a related integral
  
  • P. J. de Doelder, J. Comput. Appl. Math. 11, 325 (1984)
• Efficient Calculation of Clausen's Integral
  
  • Van E. Wood, Mathematics of Computation, Vol. 22, No. 104 (Oct., 1968), pp. 883-884

• 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
• Mathematical Moments from the AMS
• BetterExplained