"로그감마 함수"의 두 판 사이의 차이

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==개요==
 
==개요==
  
 
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==후르비츠 제타함수==
 
==후르비츠 제타함수==
  
*  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><br>
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*  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> 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><br>http://dlmf.nist.gov/5/9/ 참고<br>
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*  Binet's second expression:<math>\operatorname{Re} z > 0 </math> 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 푸리에 급수==
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==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><br>
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*  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==
  
* [[로그감마 함수]]의 테일러 급수 (http://www.wolframalpha.com/input/?i=taylor+series+of+log+gamma(1%2Bx)+at+x%3D0):<math>\log\Gamma(1+x) =-\gamma x+\sum_{k=2}^{\infty}(-1)^k \frac{\zeta(k)}{k}x^k</math><br>
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* [[로그감마 함수]]의 테일러 급수 (http://www.wolframalpha.com/input/?i=taylor+series+of+log+gamma(1%2Bx)+at+x%3D0):<math>\log\Gamma(1+x) =-\gamma x+\sum_{k=2}^{\infty}(-1)^k \frac{\zeta(k)}{k}x^k</math>
* [[정수에서의 리만제타함수의 값]]<br>
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* [[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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A는 [[Glaisher–Kinkelin 상수]]
  
 
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==스털링 공식==
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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 형태의 자료==
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==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==
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* 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.
 
* 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>
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* 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.
** Connon, Donal F, 2009
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* 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
* [http://www.math.tulane.edu/~vhm/papers_html/log-gamma.pdf INTEGRALS OF POWERS OF LOGGAMMA]<br>
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* 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
** TEWODROS AMDEBERHAN, MARK W. COFFEY, OLIVIER ESPINOSA, CHRISTOPH KOUTSCHAN, DANTE V. MANNA, AND VICTOR H. MOLL
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* 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.
* [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
 

2020년 12월 28일 (월) 03:17 기준 최신판

개요

후르비츠 제타함수



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



정적분

\(\int_{0}^{1}\log\Gamma(x)\,dx=\log\sqrt{2\pi}\)


\(\int_{0}^{\frac{1}{2}}\log\Gamma(x+1)\,dx=-\frac{1}{2}-\frac{7}{24}\log 2+\frac{1}{4}\log \pi+\frac{3}{2}\log A\)

A는 Glaisher–Kinkelin 상수



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.