적분의 주제들
이 항목의 스프링노트 원문주소
==개요
==관련된 학부 과목과 미리 알고 있으면 좋은 것들
==부정적분의 기술
==다양한 정적분의 계산
- 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)
\(\int_{\pi/4}^{\pi/2} \ln \ln \tan x\, dx=\frac{\pi}{2}\ln{\frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})}\sqrt{2\pi}\)
\(\int_0^{\infty}\frac{\log^2 x}{1+x^2} dx = \frac{ \pi^3}{8}\) - 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
\(\int_{0}^{\infty}\frac{\ln(x^{2}+1)}{x^{2}+1}\,dx=\pi\ln2\)
Gradshteyn and Ryzhik
- http://www.math.tulane.edu/~vhm/Table.html
- Directory of notes on the integrals in GR
- http://arxiv.org/find/math/1/au:+Moll_V/0/1/0/all/0/1
- The integrals in Gradshteyn and Ryzhik. Part 1: A family of logarithmic integrals.
- Victor H. Moll
- Victor H. Moll
하위페이지
==메모
- http://www.strw.leidenuniv.nl/~mathar/public/mathar20071105.pdf
- http://crd.lbl.gov/~dhbailey/expmath/maa-course/Moll-MAA.pdf
==관련된 대학원 과목
- differential Galois theory
==관련된 항목들
==관련도서
- Handbook of Integration
- Daniel Zwillinger, 1992
- 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'
==관련웹페이지
==관련논문
- Seized Opportunities
- Victor H. Moll, Notices of the AMS, Apr. 2010
- On Some Integrals Involving the Hurwitz Zeta Function: Part 1
- Olivier Espinosa and Victor H. Moll
- On Some Integrals Involving the Hurwitz Zeta Function: Part 2
- Olivier Espinosa and Victor H. Moll
- The Evaluation of Integrals: A Personal Story
- Victor Moll, Notices Amer. Math. Soc. 49(3) (2002) 311-317
- The Evolution of Integration
- A. Shenitzer and J. Steprans, The American Mathematical Monthly, Vol. 101, No. 1 (Jan., 1994), pp. 66-72
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