"더블감마함수와 반스(Barnes) G-함수"의 두 판 사이의 차이

수학노트
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(같은 사용자의 중간 판 17개는 보이지 않습니다)
1번째 줄: 1번째 줄:
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==이 항목의 스프링노트 원문주소==
 
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==개요==
 
==개요==
  
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*  더블 감마함수의 역수로 정의되는 함수<br>
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*  더블 감마함수의 역수로 정의되는 함수
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*  성질<br><math>G(1)=1</math><br><math>G(s+1) =\Gamma(s)G(s)</math><br>
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*  성질:<math>G(1)=1</math>:<math>G(s+1) =\Gamma(s)G(s)</math>
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*  자연수 n에 대하여 다음이 성립한다<br><math>G(n)=(n-1)!\times (n-2)! \times\cdots 2!\times 1!</math><br>
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*  자연수 n에 대하여 다음이 성립한다:<math>G(n)=(n-1)!\times (n-2)! \times\cdots 2!\times 1!</math>
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==근사식==
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<math>\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)</math>
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==점근급수==
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* [[점근 급수(asymptotic series)]]
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:<math>\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)</math>
  
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여기서 A는 [[Glaisher–Kinkelin 상수]] <math>A= e^{\frac{1}{12}-\zeta^\prime(-1)}= 1.28242712\dots</math>
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여기서 A는 [[Glaisher–Kinkelin 상수]] <math>A= e^{\frac{1}{12}-\zeta^\prime(-1)}= 1.28242712\dots</math>
  
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* [[스털링 공식]]<br>
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* [[스털링 공식]]
  
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==special values==
 
==special values==
  
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*  A는 [[Glaisher–Kinkelin 상수]]<br><math>G(\frac{1}{2})=2^{\frac{1}{24}}\cdot \pi^{-\frac{1}{4}}\cdot e^{\frac{1}{8}}\cdot A^{-\frac{3}{2}}</math><br><math>G(\frac{3}{4})=2^{-\frac{1}{8}}\cdot \pi^{-\frac{1}{4}}\cdot e^{\frac{1}{8}}\cdot A^{-\frac{3}{2}}</math> 또는 <math>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}}</math><br>
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*  A는 [[Glaisher–Kinkelin 상수]]:<math>G(\frac{1}{2})=2^{\frac{1}{24}}\cdot \pi^{-\frac{1}{4}}\cdot e^{\frac{1}{8}}\cdot A^{-\frac{3}{2}}</math>:<math>G(\frac{3}{4})=2^{-\frac{1}{8}}\cdot \pi^{-\frac{1}{4}}\cdot e^{\frac{1}{8}}\cdot A^{-\frac{3}{2}}</math> 또는 <math>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}}</math>
  
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==로그 삼각함수 적분과의 관계==
 
==로그 삼각함수 적분과의 관계==
  
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<math>\int_{0}^{t}\pi u \cot \pi u\,du=t\log {2\pi}+\log \frac{G(1-t)}{G(1+t)}</math>
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:<math>\int_{0}^{t}\pi u \cot \pi u\,du=t\log {2\pi}+\log \frac{G(1-t)}{G(1+t)}</math>
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:<math>\int_{0}^{t}\log(\sin \pi u)\,du=t\log(\frac{\sin \pi t}{2\pi})+\log \frac{G(1+t)}{G(1-t)}</math>
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<math>\int_{0}^{t}\log(\sin \pi u)\,du=t\log(\frac{\sin \pi t}{2\pi})+\log \frac{G(1+t)}{G(1-t)}</math>
 
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==재미있는 사실==
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* Math Overflow http://mathoverflow.net/search?q=
 
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* 네이버 지식인 http://kin.search.naver.com/search.naver?where=kin_qna&query=
 
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==역사==
 
==역사==
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* [[수학사 연표]]
  
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* http://www.google.com/search?hl=en&tbs=tl:1&q=
 
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* [[수학사연표 (역사)|수학사연표]]
 
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==메모==
 
==메모==
  
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==관련된 항목들==
 
==관련된 항목들==
  
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* [[감마함수]]<br>
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* [[감마함수]]
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* [[멀티 감마함수(multiple gamma function)]]<br>
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* [[멀티 감마함수(multiple gamma function)]]
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* [[로그 사인 적분 (log sine integrals)]]<br>
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* [[로그 사인 적분 (log sine integrals)]]
  
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==수학용어번역==
 
==수학용어번역==
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* {{학술용어집|url=hyperfactorial}}
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* 발음사전 http://www.forvo.com/search/Barnes
  
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* 단어사전 http://www.google.com/dictionary?langpair=en|ko&q=hyperfactorial
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* 발음사전 http://www.forvo.com/search/Barnes
 
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* [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= 대한수학회 수학 학술 용어집]<br>
 
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** http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=hyperfactorial
 
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** http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
 
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* [http://www.nktech.net/science/term/term_l.jsp?l_mode=cate&s_code_cd=MA 남·북한수학용어비교]
 
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* [http://kms.or.kr/home/kor/board/bulletin_list_subject.asp?bulletinid=%7BD6048897-56F9-43D7-8BB6-50B362D1243A%7D&boardname=%BC%F6%C7%D0%BF%EB%BE%EE%C5%E4%B7%D0%B9%E6&globalmenu=7&localmenu=4 대한수학회 수학용어한글화 게시판]
 
  
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==사전 형태의 자료==
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==사전 형태의 자료==
 
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* http://ko.wikipedia.org/wiki/
 
 
* http://en.wikipedia.org/wiki/Barnes_G-function
 
* http://en.wikipedia.org/wiki/Barnes_G-function
 
* http://www.wolframalpha.com/input/?i=Barnes+G-function
 
* http://www.wolframalpha.com/input/?i=Barnes+G-function
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* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]<br>
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* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
 
** [http://dlmf.nist.gov/5.17 § 5.17. Barnes’ -Function (Double Gamma Function)]
 
** [http://dlmf.nist.gov/5.17 § 5.17. Barnes’ -Function (Double Gamma Function)]
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* [http://www.research.att.com/~njas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]<br>
 
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** http://www.research.att.com/~njas/sequences/?q=
 
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==관련논문==
 
==관련논문==
  
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* [http://www.cs.cmu.edu/~adamchik/articles/Srivastava/ch_sr.pdf Multiple Gamma and Related Functions]<br>
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* [http://www.cs.cmu.edu/~adamchik/articles/Srivastava/ch_sr.pdf Multiple Gamma and Related Functions]
 
** J. Choi, H. M. Srivastava, V.S. Adamchik , Applied Mathematics and Computation, 134 (2003), 515-533
 
** J. Choi, H. M. Srivastava, V.S. Adamchik , Applied Mathematics and Computation, 134 (2003), 515-533
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* [http://projecteuclid.org/euclid.tjm/1270472992 A Proof of the Classical Kronecker Limit Formula]<br>
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* [http://projecteuclid.org/euclid.tjm/1270472992 A Proof of the Classical Kronecker Limit Formula]
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**  Takuro SHINTANI. Source: Tokyo J. of Math. Volume 03, Number 2 (1980), 191-199<br>
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**  Takuro SHINTANI. Source: Tokyo J. of Math. Volume 03, Number 2 (1980), 191-199
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* Barnes, E. W. 2013. “The Genesis of the Double Gamma Functions.” Proceedings of the London Mathematical Society S1-31 (1): 358. doi:10.1112/plms/s1-31.1.358.
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* http://www.jstor.org/action/doBasicSearch?Query=
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[[분류:특수함수]]
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* http://www.ams.org/mathscinet
 
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* http://dx.doi.org/
 
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==블로그==
 
  
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*  구글 블로그 검색<br>
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==메타데이터==
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** http://blogsearch.google.com/blogsearch?q=
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===위키데이터===
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* [http://navercast.naver.com/science/list 네이버 오늘의과학]
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* ID : [https://www.wikidata.org/wiki/Q808463 Q808463]
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===Spacy 패턴 목록===
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* [{'LOWER': 'barnes'}, {'LOWER': 'g'}, {'OP': '*'}, {'LEMMA': 'function'}]

2021년 2월 17일 (수) 04:02 기준 최신판

개요

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



점근급수

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

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



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

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





역사


메모

관련된 항목들



수학용어번역


사전 형태의 자료


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
  • Barnes, E. W. 2013. “The Genesis of the Double Gamma Functions.” Proceedings of the London Mathematical Society S1-31 (1): 358. doi:10.1112/plms/s1-31.1.358.

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'barnes'}, {'LOWER': 'g'}, {'OP': '*'}, {'LEMMA': 'function'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=더블감마함수와_반스(Barnes)_G-함수&oldid=52228"