"적분의 주제들"의 두 판 사이의 차이

2015년 8월 3일 (월) 21:42 판

로그 사인 적분 (log sine integrals) \[\int_{0}^{\pi/2}\log^2(\sin x)\,dx=\frac{\pi}{2}(\log 2)^2+\frac{\pi^3}{24}\]
로그 탄젠트 적분(log tangent integral) \[\int_{\pi/4}^{\pi/2} \ln \ln \tan x\, dx=\frac{\pi}{2}\ln \left(\frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})}\sqrt{2\pi}\right)\] \[\int_0^{\infty}\frac{\log^2 x}{1+x^2} dx = \frac{ \pi^3}{8}\]
다이로그 함수(dilogarithm ) \[\int_{0}^{\infty}\log(1+e^{-x})\,dx=\frac{\pi^2}{12}\]
가우시안 적분 \[\int_{-\infty}^{\infty}e^{-\frac{x^2}{2}}dx=\sqrt{2\pi}\]
라마누잔의 정적분 \[\int_{0}^{\infty}\frac{x e^{-\sqrt{5}x}}{\cosh{x}}\,dx=\frac{1}{8}(\psi^{(1)}(\frac{1+\sqrt{5}}{4})-\psi^{(1)}(\frac{3+\sqrt{5}}{4}))\]
로그함수와 유리함수가 있는 정적분 \[\int_{0}^{\infty}\frac{\ln(x^{2}+1)}{x^{2}+1}\,dx=\pi\ln2\]

http://www.strw.leidenuniv.nl/~mathar/public/mathar20071105.pdf
http://crd.lbl.gov/~dhbailey/expmath/maa-course/Moll-MAA.pdf
Scarpello, Giovanni Mingari, and Daniele Ritelli. “New Hypergeometric Formulae to $\pi$ Arising from M. Roberts Hyperelliptic Reductions.” arXiv:1507.06681 [math], July 23, 2015. http://arxiv.org/abs/1507.06681.

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 ==관련논문==
+* Gonzalez, Ivan, Lin Jiu, and Victor H. Moll. “Pochhammer Symbol with Negative Indices. A New Rule for the Method of Brackets.” arXiv:1508.00056 [math], July 31, 2015. http://arxiv.org/abs/1508.00056.
 * Ahmed, Zafar. ‘Ahmed’s Integral: The Maiden Solution’. arXiv:1411.5169 Null, 19 November 2014. http://arxiv.org/abs/1411.5169.
 * [http://www.ams.org/notices/201004/rtx100400476p.pdf Seized Opportunities]