"적분의 주제들"의 두 판 사이의 차이

2015년 3월 28일 (토) 05:15 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 \[\int_{0}^{\infty}\frac{\ln(x^{2}+1)}{x^{2}+1}\,dx=\pi\ln2\]

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 ==관련도서==
+* Nahin, Paul J. Inside Interesting Integrals: A Collection of Sneaky Tricks, Sly Substitutions, and Numerous Other Stupendously Clever, Awesomely Wicked, and ... 2015 edition. New York: Springer, 2014.
-* [http://books.google.com/books?id=DQd4wfV7fo0C Handbook of Integration]
+* Zwillinger, Daniel. The Handbook of Integration. Taylor & Francis, 1992.  http://books.google.com/books?id=DQd4wfV7fo0C
-**  Daniel Zwillinger, 1992
+* Boros, George, and Victor Moll. Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals. 1 edition. Cambridge, UK ; New York: Cambridge University Press, 2004. http://www.amazon.com/Irresistible-Integrals-Symbolics-Experiments-Evaluation/dp/0521796369
-* [http://www.amazon.com/Irresistible-Integrals-Symbolics-Experiments-Evaluation/dp/0521796369 Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals]
-** George Boros and Victor Moll, 2004
 * [http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=Galois%27 http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=Galois']
 ==관련웹페이지==