로그감마 함수

 

 

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

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

 

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

A는 Glaisher–Kinkelin 상수

 

 

스털링 공식

• 스털링 공식

 

 

 

 

 

관련된 항목들

• 후르비츠 제타함수(Hurwitz zeta function)

 

   

사전 형태의 자료

• http://mathworld.wolfram.com/LogGammaFunction.html
• http://www.wolframalpha.com/input/?i=Loggamma

 

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.
• 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.