"더블감마함수와 반스(Barnes) G-함수"의 두 판 사이의 차이
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개요==
special values==
관련논문==
Pythagoras0 (토론 | 기여) 잔글 (찾아 바꾸기 – “</h5>” 문자열을 “==” 문자열로) |
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| − | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">이 항목의 스프링노트 원문주소 | + | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">이 항목의 스프링노트 원문주소== |
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| − | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">개요 | + | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">개요== |
* 더블 감마함수의 역수로 정의되는 함수<br> | * 더블 감마함수의 역수로 정의되는 함수<br> | ||
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| − | <h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">근사식 | + | <h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">근사식== |
<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> | <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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| − | <h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">special values | + | <h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">special values== |
* 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> | * 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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| − | <h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">로그 삼각함수 적분과의 관계 | + | <h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">로그 삼각함수 적분과의 관계== |
<math>\int_{0}^{t}\pi u \cot \pi u\,du=t\log {2\pi}+\log \frac{G(1-t)}{G(1+t)}</math> | <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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| − | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">관련된 항목들 | + | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">관련된 항목들== |
* [[감마함수]]<br> | * [[감마함수]]<br> | ||
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| − | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">수학용어번역 | + | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">수학용어번역== |
* 단어사전 http://www.google.com/dictionary?langpair=en|ko&q=hyperfactorial | * 단어사전 http://www.google.com/dictionary?langpair=en|ko&q=hyperfactorial | ||
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* http://ko.wikipedia.org/wiki/ | * http://ko.wikipedia.org/wiki/ | ||
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* [http://www.cs.cmu.edu/~adamchik/articles/Srivastava/ch_sr.pdf Multiple Gamma and Related Functions]<br> | * [http://www.cs.cmu.edu/~adamchik/articles/Srivastava/ch_sr.pdf Multiple Gamma and Related Functions]<br> | ||
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* 도서내검색<br> | * 도서내검색<br> | ||
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* 네이버 뉴스 검색 (키워드 수정)<br> | * 네이버 뉴스 검색 (키워드 수정)<br> | ||
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| − | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">블로그 | + | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">블로그== |
* 구글 블로그 검색<br> | * 구글 블로그 검색<br> | ||
2012년 11월 1일 (목) 12:14 판
이 항목의 스프링노트 원문주소==
개요==
- 더블 감마함수의 역수로 정의되는 함수
- 성질
\(G(1)=1\)
\(G(s+1) =\Gamma(s)G(s)\)
- 자연수 n에 대하여 다음이 성립한다
\(G(n)=(n-1)!\times (n-2)! \times\cdots 2!\times 1!\)
\(G(1)=1\)
\(G(s+1) =\Gamma(s)G(s)\)
\(G(n)=(n-1)!\times (n-2)! \times\cdots 2!\times 1!\)
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 \(A= e^{\frac{1}{12}-\zeta^\prime(-1)}= 1.28242712\dots\)
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 \(G(\frac{3}{4})=2^{-\frac{1}{8}}\cdot \pi^{-\frac{1}{4}}\cdot e^{\frac{3}{32}+\frac{G}{4\pi}}\cdot A^{-\frac{9}{8}}\cdot \Gamma(\frac{1}{4})^{\frac{1}{4}}\)
\(G(\frac{1}{2})=2^{\frac{1}{24}}\cdot \pi^{-\frac{1}{4}}\cdot e^{\frac{1}{8}}\cdot A^{-\frac{3}{2}}\)
\(G(\frac{3}{4})=2^{-\frac{1}{8}}\cdot \pi^{-\frac{1}{4}}\cdot e^{\frac{1}{8}}\cdot A^{-\frac{3}{2}}\) 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==
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
- J. Choi, H. M. Srivastava, V.S. Adamchik , Applied Mathematics and Computation, 134 (2003), 515-533
- Takuro SHINTANI. Source: Tokyo J. of Math. Volume 03, Number 2 (1980), 191-199