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Pythagoras0 (토론 | 기여) |
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| (사용자 2명의 중간 판 36개는 보이지 않습니다) | |||
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| − | + | ==관련된 학부 과목과 미리 알고 있으면 좋은 것들== | |
| − | + | * [[일변수미적분학]] | |
| + | * [[복소함수론]] | ||
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| + | ==부정적분의 기술== | ||
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| + | * [[오일러 치환]] | ||
| + | * [[삼각치환]] | ||
| + | * [[다이로그 함수와 부정적분]] | ||
| + | * [[역함수를 이용한 치환적분]] | ||
| + | * [[부정적분의 초등함수 표현(Integration in finite terms)]] | ||
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| − | + | ==다양한 정적분의 계산== | |
| − | * [[ | + | * [[로그 사인 적분 (log sine integrals)]] :<math>\int_{0}^{\pi/2}\log^2(\sin x)\,dx=\frac{\pi}{2}(\log 2)^2+\frac{\pi^3}{24}</math> |
| − | * [[ | + | * [[로그 탄젠트 적분(log tangent integral)]] :<math>\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)</math> :<math>\int_0^{\infty}\frac{\log^2 x}{1+x^2} dx = \frac{ \pi^3}{8}</math> |
| + | * [[다이로그 함수(dilogarithm)|다이로그 함수(dilogarithm )]] :<math>\int_{0}^{\infty}\log(1+e^{-x})\,dx=\frac{\pi^2}{12}</math> | ||
| + | * [[가우시안 적분]] :<math>\int_{-\infty}^{\infty}e^{-\frac{x^2}{2}}dx=\sqrt{2\pi}</math> | ||
| + | * [[라마누잔의 정적분]] :<math>\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}))</math> | ||
| + | * [[로그함수와 유리함수가 있는 정적분]] :<math>\int_{0}^{\infty}\frac{\ln(x^{2}+1)}{x^{2}+1}\,dx=\pi\ln2</math> | ||
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| − | + | ==Gradshteyn and Ryzhik== | |
| − | * [[ | + | * [http://www.math.tulane.edu/%7Evhm/Table.html http://www.math.tulane.edu/~vhm/Table.html] |
| + | * [http://www.math.tulane.edu/%7Evhm/web_html/directory.html Directory of notes on the integrals in GR] | ||
| + | * http://arxiv.org/find/math/1/au:+Moll_V/0/1/0/all/0/1 | ||
| + | * [http://arxiv.org/abs/0704.3872v2 The integrals in Gradshteyn and Ryzhik. Part 1: A family of logarithmic integrals.] | ||
| + | ** Victor H. Moll | ||
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| − | + | ==메모== | |
| − | + | * [http://www.strw.leidenuniv.nl/%7Emathar/public/mathar20071105.pdf http://www.strw.leidenuniv.nl/~mathar/public/mathar20071105.pdf] | |
| + | * [http://crd.lbl.gov/%7Edhbailey/expmath/maa-course/Moll-MAA.pdf http://crd.lbl.gov/~dhbailey/expmath/maa-course/Moll-MAA.pdf] | ||
| + | * Scarpello, Giovanni Mingari, and Daniele Ritelli. “New Hypergeometric Formulae to <math>\pi</math> 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 | * differential Galois theory | ||
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| − | + | ==관련된 항목들== | |
| − | * [[ | + | * [[타원적분]] |
| − | + | * [[오일러 치환]] | |
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| − | * [[오일러 치환 | ||
* [[삼각치환]] | * [[삼각치환]] | ||
| + | * [[감마함수]] | ||
| + | * [[다이로그 함수(dilogarithm)]] | ||
| + | * [[L-함수, 제타함수와 디리클레 급수]] | ||
| + | * [[초기하급수(Hypergeometric series)]] | ||
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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. | ||
| + | * 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%27 http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=Galois'] | ||
| − | + | ==관련웹페이지== | |
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| − | + | * http://integrals.wolfram.com/index.jsp | |
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| − | + | ==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. | ||
| + | * [http://www.ams.org/notices/201004/rtx100400476p.pdf Seized Opportunities] | ||
| + | ** Victor H. Moll, Notices of the AMS, Apr. 2010 | ||
| + | * [http://www.springerlink.com/content/wql8d40h20jljxp2/ On Some Integrals Involving the Hurwitz Zeta Function: Part 1] | ||
| + | ** Olivier Espinosa and Victor H. Moll | ||
| + | * [http://www.springerlink.com/content/t285842772wv0767/ On Some Integrals Involving the Hurwitz Zeta Function: Part 2] | ||
| + | ** Olivier Espinosa and Victor H. Moll | ||
| + | * Victor Moll, [http://www.math.tulane.edu/%7Evhm/papers_html/fea-moll.pdf 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. | ||
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| − | + | [[분류:적분]] | |
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2020년 11월 13일 (금) 07:15 기준 최신판
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) \[\int_{\pi/4}^{\pi/2} \ln \ln \tan x\, dx=\frac{\pi}{2}\ln \left(\frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})}\sqrt{2\pi}\right)\] \[\int_0^{\infty}\frac{\log^2 x}{1+x^2} dx = \frac{ \pi^3}{8}\]
- 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
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
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.