적분의 주제들

관련된 학부 과목과 미리 알고 있으면 좋은 것들

• 일변수미적분학
• 복소함수론

부정적분의 기술

• 오일러 치환
• 삼각치환
• 다이로그 함수와 부정적분
• 역함수를 이용한 치환적분
• 부정적분의 초등함수 표현(Integration in finite terms)

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) \[\int_{\pi/4}^{\pi/2} \ln \ln \tan x\, dx=\frac{\pi}{2}\ln \left(\frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})}\sqrt{2\pi}\right)\] \[\int_0^{\infty}\frac{\log^2 x}{1+x^2} dx = \frac{ \pi^3}{8}\]
• 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 \[\int_{0}^{\infty}\frac{\ln(x^{2}+1)}{x^{2}+1}\,dx=\pi\ln2\]

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

메모

• 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.

관련된 대학원 과목

• differential Galois theory

관련된 항목들

• 타원적분
• 오일러 치환
• 삼각치환
• 감마함수
• 다이로그 함수(dilogarithm)
• L-함수, 제타함수와 디리클레 급수
• 초기하급수(Hypergeometric series)

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.
• Zwillinger, Daniel. The Handbook of Integration. Taylor & Francis, 1992. http://books.google.com/books?id=DQd4wfV7fo0C
• 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/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=Galois'

관련웹페이지

• http://integrals.wolfram.com/index.jsp

리뷰, 에세이, 강의노트

• 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.
• 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
• Victor Moll, The Evaluation of Integrals: A Personal Story, Notices Amer. Math. Soc. 49(3) (2002) 311-317
• Shenitzer, A., and J. Steprans. “The Evolution of Integration.” The American Mathematical Monthly 101, no. 1 (1994): 66–72. doi:10.2307/2325128. http://www.jstor.org/stable/2325128
• Abramowitz, Milton. “On the Practical Evaluation of Integrals.” Society for Industrial and Applied Mathematics. Journal of the Society of Industrial and Applied Mathematics 2, no. 1 (March 1954): 16. doi:http://dx.doi.org/10.1137/0102003.