"로그감마 함수"의 두 판 사이의 차이
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==개요== | ==개요== | ||
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* [[후르비츠 제타함수(Hurwitz zeta function)]] | * [[후르비츠 제타함수(Hurwitz zeta function)]] | ||
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==사전 형태의 자료== | ==사전 형태의 자료== | ||
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* 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== | ||
| − | + | * 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. | |
* [http://arxiv.org/abs/0903.4323 Fourier series representations of the logarithms of the Euler gamma function and the Barnes multiple gamma functions]<br> | * [http://arxiv.org/abs/0903.4323 Fourier series representations of the logarithms of the Euler gamma function and the Barnes multiple gamma functions]<br> | ||
** Connon, Donal F, 2009 | ** Connon, Donal F, 2009 | ||
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* [http://www.math.titech.ac.jp/~tosho/Preprints/pdf/128.pdf Kummer's Formula for Multiple Gamma Functions]<br> | * [http://www.math.titech.ac.jp/~tosho/Preprints/pdf/128.pdf Kummer's Formula for Multiple Gamma Functions]<br> | ||
** Shin-ya Koyama, Nobushige Kurokawa, 2002 | ** Shin-ya Koyama, Nobushige Kurokawa, 2002 | ||
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2014년 4월 11일 (금) 07:12 판
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
- 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.
- Fourier series representations of the logarithms of the Euler gamma function and the Barnes multiple gamma functions
- Connon, Donal F, 2009
- INTEGRALS OF POWERS OF LOGGAMMA
- TEWODROS AMDEBERHAN, MARK W. COFFEY, OLIVIER ESPINOSA, CHRISTOPH KOUTSCHAN, DANTE V. MANNA, AND VICTOR H. MOLL
- Kummer's Formula for Multiple Gamma Functions
- Shin-ya Koyama, Nobushige Kurokawa, 2002