더블감마함수와 반스(Barnes) G-함수

 

 

개요

• 더블 감마함수의 역수로 정의되는 함수
  
• 성질
  \(G(1)=1\)
  \(G(s+1) =\Gamma(s)G(s)\)
  
• 자연수 n에 대하여 다음이 성립한다
  \(G(n)=(n-1)!\times (n-2)! \times\cdots 2!\times 1!\)
  

 

 

근사식

\(\log G(z+1)=\frac{1}{12}~-~\log A~+~\frac{z}{2}\log 2\pi~+~\left(\frac{z^2}{2} -\frac{1}{12}\right)\log z~-~\frac{3z^2}{4}~+~ \sum_{k=1}^{N}\frac{B_{2k + 2}}{4k\left(k + 1\right)z^{2k}}~+~O\left(\frac{1}{z^{2N + 2}}\right)\)

여기서 A는 Glaisher–Kinkelin 상수 \(A= e^{\frac{1}{12}-\zeta^\prime(-1)}= 1.28242712\dots\)

• 스털링 공식
  

 

 

special values

• A는 Glaisher–Kinkelin 상수
  \(G(\frac{1}{2})=2^{\frac{1}{24}}\cdot \pi^{-\frac{1}{4}}\cdot e^{\frac{1}{8}}\cdot A^{-\frac{3}{2}}\)
  \(G(\frac{3}{4})=2^{-\frac{1}{8}}\cdot \pi^{-\frac{1}{4}}\cdot e^{\frac{1}{8}}\cdot A^{-\frac{3}{2}}\) 또는 \(G(\frac{3}{4})=2^{-\frac{1}{8}}\cdot \pi^{-\frac{1}{4}}\cdot e^{\frac{3}{32}+\frac{G}{4\pi}}\cdot A^{-\frac{9}{8}}\cdot \Gamma(\frac{1}{4})^{\frac{1}{4}}\)
  

 

 

로그 삼각함수 적분과의 관계

\(\int_{0}^{t}\pi u \cot \pi u\,du=t\log {2\pi}+\log \frac{G(1-t)}{G(1+t)}\)

\(\int_{0}^{t}\log(\sin \pi u)\,du=t\log(\frac{\sin \pi t}{2\pi})+\log \frac{G(1+t)}{G(1-t)}\)

 

 

 

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
•  

 

 

 

 

관련된 항목들

• 로그 사인 적분 (log sine integrals)
  

 

 

수학용어번역

• 단어사전 http://www.google.com/dictionary?langpair=en%7Cko&q=hyperfactorial
• 발음사전 http://www.forvo.com/search/Barnes
• 대한수학회 수학 학술 용어집
  
  • http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=hyperfactorial
  • 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/Barnes_G-function
• http://www.wolframalpha.com/input/?i=Barnes+G-function
• NIST Digital Library of Mathematical Functions
  
  • § 5.17. Barnes’ -Function (Double Gamma Function)
• The On-Line Encyclopedia of Integer Sequences
  
  • http://www.research.att.com/~njas/sequences/?q=

 

 

관련논문

• Multiple Gamma and Related Functions
  
  • J. Choi, H. M. Srivastava, V.S. Adamchik , Applied Mathematics and Computation, 134 (2003), 515-533
• A Proof of the Classical Kronecker Limit Formula
  
  • Takuro SHINTANI. Source: Tokyo J. of Math. Volume 03, Number 2 (1980), 191-199
    

• 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