"로그감마 함수"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) |
Pythagoras0 (토론 | 기여) |
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==개요== | ==개요== | ||
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==후르비츠 제타함수== | ==후르비츠 제타함수== | ||
| − | * Lerch의 공식 : | + | * Lerch의 공식 : [[후르비츠 제타함수(Hurwitz zeta function)]]의 미분:<math>\frac{\partial }{\partial s}\zeta(s,a)|_{s=0} =\log \frac{\Gamma(a)}{\sqrt{2\pi}}</math> |
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==적분표현== | ==적분표현== | ||
| − | * Binet's second expression:<math>\operatorname{Re} z > 0 </math> | + | * Binet's second expression:<math>\operatorname{Re} z > 0 </math> 일 때, <math>\log \Gamma(z)=(z-\frac{1}{2})\log z -z+\frac{1}{2}\log 2\pi+ 2\int_0^{\infty}\frac{\tan^{-1}(t/z)}{e^{2\pi t} -1}dt</math>http://dlmf.nist.gov/5/9/ 참고 |
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| − | ==쿰머의 | + | ==쿰머의 푸리에 급수== |
* 쿰머 (1847):<math>\begin{eqnarray}\log\Gamma(x)=\log\sqrt{2\pi}-\frac{1}{2}\log(2\sin\pi x)+\frac{1}{2}(\gamma+2\log\sqrt{2\pi})(1-2x)+\frac{1}{\pi}\sum_{k=1}^{\infty}\frac{\log k}{k}\sin 2\pi kx \nonumber \\ =(\frac{1}{2}-x)(\gamma+\log 2)+(1-x)\log \pi -\frac{1}{2}\log(\sin\pi x)+\frac{1}{\pi}\sum_{k=1}^{\infty}\frac{\log k}{k}\sin 2\pi kx \nonumber \end{eqnarray} </math> | * 쿰머 (1847):<math>\begin{eqnarray}\log\Gamma(x)=\log\sqrt{2\pi}-\frac{1}{2}\log(2\sin\pi x)+\frac{1}{2}(\gamma+2\log\sqrt{2\pi})(1-2x)+\frac{1}{\pi}\sum_{k=1}^{\infty}\frac{\log k}{k}\sin 2\pi kx \nonumber \\ =(\frac{1}{2}-x)(\gamma+\log 2)+(1-x)\log \pi -\frac{1}{2}\log(\sin\pi x)+\frac{1}{\pi}\sum_{k=1}^{\infty}\frac{\log k}{k}\sin 2\pi kx \nonumber \end{eqnarray} </math> | ||
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==테일러 급수== | ==테일러 급수== | ||
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* [[정수에서의 리만제타함수의 값]] | * [[정수에서의 리만제타함수의 값]] | ||
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==정적분== | ==정적분== | ||
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<math>\int_{0}^{1}\log\Gamma(x)\,dx=\log\sqrt{2\pi}</math> | <math>\int_{0}^{1}\log\Gamma(x)\,dx=\log\sqrt{2\pi}</math> | ||
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<math>\int_{0}^{\frac{1}{2}}\log\Gamma(x+1)\,dx=-\frac{1}{2}-\frac{7}{24}\log 2+\frac{1}{4}\log \pi+\frac{3}{2}\log A</math> | <math>\int_{0}^{\frac{1}{2}}\log\Gamma(x+1)\,dx=-\frac{1}{2}-\frac{7}{24}\log 2+\frac{1}{4}\log \pi+\frac{3}{2}\log A</math> | ||
| − | + | A는 [[Glaisher–Kinkelin 상수]] | |
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| − | == | + | ==스털링 공식== |
* [[스털링 공식]] | * [[스털링 공식]] | ||
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==메모== | ==메모== | ||
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==관련된 항목들== | ==관련된 항목들== | ||
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* [[후르비츠 제타함수(Hurwitz zeta function)]] | * [[후르비츠 제타함수(Hurwitz zeta function)]] | ||
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| − | ==사전 | + | ==사전 형태의 자료== |
* http://mathworld.wolfram.com/LogGammaFunction.html | * http://mathworld.wolfram.com/LogGammaFunction.html | ||
* http://www.wolframalpha.com/input/?i=Loggamma | * http://www.wolframalpha.com/input/?i=Loggamma | ||
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==관련논문== | ==관련논문== | ||
2020년 12월 28일 (월) 03:17 기준 최신판
개요
후르비츠 제타함수
- Lerch의 공식 : 후르비츠 제타함수(Hurwitz zeta function)의 미분\[\frac{\partial }{\partial s}\zeta(s,a)|_{s=0} =\log \frac{\Gamma(a)}{\sqrt{2\pi}}\]
적분표현
- Binet's second expression\[\operatorname{Re} z > 0 \] 일 때, \(\log \Gamma(z)=(z-\frac{1}{2})\log z -z+\frac{1}{2}\log 2\pi+ 2\int_0^{\infty}\frac{\tan^{-1}(t/z)}{e^{2\pi t} -1}dt\)http://dlmf.nist.gov/5/9/ 참고
쿰머의 푸리에 급수
- 쿰머 (1847)\[\begin{eqnarray}\log\Gamma(x)=\log\sqrt{2\pi}-\frac{1}{2}\log(2\sin\pi x)+\frac{1}{2}(\gamma+2\log\sqrt{2\pi})(1-2x)+\frac{1}{\pi}\sum_{k=1}^{\infty}\frac{\log k}{k}\sin 2\pi kx \nonumber \\ =(\frac{1}{2}-x)(\gamma+\log 2)+(1-x)\log \pi -\frac{1}{2}\log(\sin\pi x)+\frac{1}{\pi}\sum_{k=1}^{\infty}\frac{\log k}{k}\sin 2\pi kx \nonumber \end{eqnarray} \]
테일러 급수
- 로그감마 함수의 테일러 급수 (http://www.wolframalpha.com/input/?i=taylor+series+of+log+gamma(1%2Bx)+at+x%3D0)\[\log\Gamma(1+x) =-\gamma x+\sum_{k=2}^{\infty}(-1)^k \frac{\zeta(k)}{k}x^k\]
- 정수에서의 리만제타함수의 값
정적분
\(\int_{0}^{1}\log\Gamma(x)\,dx=\log\sqrt{2\pi}\)
\(\int_{0}^{\frac{1}{2}}\log\Gamma(x+1)\,dx=-\frac{1}{2}-\frac{7}{24}\log 2+\frac{1}{4}\log \pi+\frac{3}{2}\log A\)
스털링 공식
메모
관련된 항목들
사전 형태의 자료
관련논문
- Diamond, Harold G., and Armin Straub. “Bounds for the Logarithm of the Euler Gamma Function and Its Derivatives.” arXiv:1508.03267 [math], August 13, 2015. http://arxiv.org/abs/1508.03267.
- Kowalenko, Victor. “Exactification of Stirling’s Approximation for the Logarithm of the Gamma Function.” arXiv:1404.2705 [math], April 10, 2014. http://arxiv.org/abs/1404.2705.
- Connon, Donal F. “Fourier Series Representations of the Logarithms of the Euler Gamma Function and the Barnes Multiple Gamma Functions.” arXiv:0903.4323 [math], March 25, 2009. http://arxiv.org/abs/0903.4323.
- Amdeberhan, Tewodros, Mark W. Coffey, Olivier Espinosa, Christoph Koutschan, Dante V. Manna, and Victor H. Moll. “Integrals of Powers of Loggamma.” Proceedings of the American Mathematical Society 139, no. 2 (2011): 535–45. doi:10.1090/S0002-9939-2010-10589-0. ,http://www.math.tulane.edu/~vhm/papers_html/log-gamma.pdf
- Koyama, Shin-ya, and Nobushige Kurokawa. "Kummer's formula for multiple gamma functions." JOURNAL-RAMANUJAN MATHEMATICAL SOCIETY 18.1 (2003): 87-107. http://www.math.titech.ac.jp/~tosho/Preprints/pdf/128.pdf
- Berndt, Bruce C. “The Gamma Function and the Hurwitz Zeta-Function.” The American Mathematical Monthly 92, no. 2 (1985): 126–30. doi:10.2307/2322640.