# 로그감마 함수

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## 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)$\begin{eqnarray}\log\Gamma(x)=\log\sqrt{2\pi}-\frac{1}{2}\log(2\sin\pi x)+\frac{1}{2}(\gamma+2\log\sqrt{2\pi})(1-2x)+\frac{1}{\pi}\sum_{k=1}^{\infty}\frac{\log k}{k}\sin 2\pi kx \nonumber \\ =(\frac{1}{2}-x)(\gamma+\log 2)+(1-x)\log \pi -\frac{1}{2}\log(\sin\pi x)+\frac{1}{\pi}\sum_{k=1}^{\infty}\frac{\log k}{k}\sin 2\pi kx \nonumber \end{eqnarray}$

## 정적분

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

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

## 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.