"적분의 주제들"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) |
Pythagoras0 (토론 | 기여) |
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| 4번째 줄: | 4번째 줄: | ||
* [[복소함수론]] | * [[복소함수론]] | ||
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==부정적분의 기술== | ==부정적분의 기술== | ||
| 12번째 줄: | 12번째 줄: | ||
* [[오일러 치환]] | * [[오일러 치환]] | ||
* [[삼각치환]] | * [[삼각치환]] | ||
| − | * [[다이로그 함수와 부정적분]] | + | * [[다이로그 함수와 부정적분]] |
| − | * [[역함수를 이용한 | + | * [[역함수를 이용한 치환적분]] |
* [[부정적분의 초등함수 표현(Integration in finite terms)]] | * [[부정적분의 초등함수 표현(Integration in finite terms)]] | ||
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==다양한 정적분의 계산== | ==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 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 | + | * [[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 | + | * [[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 | + | * [[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 | + | * [[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 | + | * [[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== | ==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/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://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/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.] | + | * [http://arxiv.org/abs/0704.3872v2 The integrals in Gradshteyn and Ryzhik. Part 1: A family of logarithmic integrals.] |
| − | ** Victor H. Moll | + | ** Victor H. Moll |
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==메모== | ==메모== | ||
| 49번째 줄: | 49번째 줄: | ||
* [http://crd.lbl.gov/%7Edhbailey/expmath/maa-course/Moll-MAA.pdf http://crd.lbl.gov/~dhbailey/expmath/maa-course/Moll-MAA.pdf] | * [http://crd.lbl.gov/%7Edhbailey/expmath/maa-course/Moll-MAA.pdf http://crd.lbl.gov/~dhbailey/expmath/maa-course/Moll-MAA.pdf] | ||
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==관련된 대학원 과목== | ==관련된 대학원 과목== | ||
| 57번째 줄: | 57번째 줄: | ||
* differential Galois theory | * differential Galois theory | ||
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==관련된 항목들== | ==관련된 항목들== | ||
| 67번째 줄: | 67번째 줄: | ||
* [[삼각치환]] | * [[삼각치환]] | ||
* [[감마함수]] | * [[감마함수]] | ||
| − | * [[다이로그 함수(dilogarithm | + | * [[다이로그 함수(dilogarithm)]] |
* [[L-함수, 제타함수와 디리클레 급수]] | * [[L-함수, 제타함수와 디리클레 급수]] | ||
* [[초기하급수(Hypergeometric series)]] | * [[초기하급수(Hypergeometric series)]] | ||
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==관련도서== | ==관련도서== | ||
| − | * [http://books.google.com/books?id=DQd4wfV7fo0C Handbook of Integration] | + | * [http://books.google.com/books?id=DQd4wfV7fo0C Handbook of Integration] |
| − | ** | + | ** Daniel Zwillinger, 1992 |
| − | * [http://www.amazon.com/Irresistible-Integrals-Symbolics-Experiments-Evaluation/dp/0521796369 Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals] | + | * [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 | ** 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'] | * [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'] | ||
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==관련웹페이지== | ==관련웹페이지== | ||
| 93번째 줄: | 93번째 줄: | ||
* http://integrals.wolfram.com/index.jsp | * http://integrals.wolfram.com/index.jsp | ||
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==관련논문== | ==관련논문== | ||
| − | * [http://www.ams.org/notices/201004/rtx100400476p.pdf Seized Opportunities] | + | * [http://www.ams.org/notices/201004/rtx100400476p.pdf Seized Opportunities] |
| − | ** Victor H. Moll, | + | ** 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] | + | * [http://www.springerlink.com/content/wql8d40h20jljxp2/ On Some Integrals Involving the Hurwitz Zeta Function: Part 1] |
| − | ** Olivier | + | ** Olivier Espinosa and Victor H. Moll |
| − | * [http://www.springerlink.com/content/t285842772wv0767/ On Some Integrals Involving the Hurwitz Zeta Function: Part 2] | + | * [http://www.springerlink.com/content/t285842772wv0767/ On Some Integrals Involving the Hurwitz Zeta Function: Part 2] |
| − | ** Olivier | + | ** Olivier Espinosa and Victor H. Moll |
| − | * [http://www.math.tulane.edu/%7Evhm/papers_html/fea-moll.pdf The Evaluation of Integrals: A Personal Story] | + | * [http://www.math.tulane.edu/%7Evhm/papers_html/fea-moll.pdf The Evaluation of Integrals: A Personal Story] |
** Victor Moll, Notices Amer. Math. Soc. 49(3) (2002) 311-317 | ** Victor Moll, Notices Amer. Math. Soc. 49(3) (2002) 311-317 | ||
| − | * [http://www.jstor.org/stable/2325128 The Evolution of Integration] | + | * [http://www.jstor.org/stable/2325128 The Evolution of Integration] |
| − | ** A. Shenitzer and J. Steprans, | + | ** A. Shenitzer and J. Steprans, The American Mathematical Monthly, Vol. 101, No. 1 (Jan., 1994), pp. 66-72 |
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==블로그== | ==블로그== | ||
* 수학과 잡담을 위한 소박한 장소 | * 수학과 잡담을 위한 소박한 장소 | ||
| − | ** [http://sos440.tistory.com/category/%EC%88%98%ED%95%99%20%EC%9E%A1%EB%8B%B4/%EC%98%A4%EB%8A%98%EC%9D%98%20%EA%B3%84%EC%82%B0 '오늘의 계산'] | + | ** [http://sos440.tistory.com/category/%EC%88%98%ED%95%99%20%EC%9E%A1%EB%8B%B4/%EC%98%A4%EB%8A%98%EC%9D%98%20%EA%B3%84%EC%82%B0 '오늘의 계산'] 카테고리 |
[[분류:적분]] | [[분류:적분]] | ||
2013년 3월 26일 (화) 02:30 판
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
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
블로그
- 수학과 잡담을 위한 소박한 장소
- '오늘의 계산' 카테고리