"오일러-가우스 초기하함수2F1"의 두 판 사이의 차이
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| 12번째 줄: | 12번째 줄: | ||
* 적분표현<br><math>\,_2F_1(a,b;c;z)=\frac{\Gamma(c)}{\Gamma(c-a)\Gamma(a)}\int_0^1t^{a-1}(1-t)^{c-a-1}(1-zt)^{-b}\,dt</math><br> | * 적분표현<br><math>\,_2F_1(a,b;c;z)=\frac{\Gamma(c)}{\Gamma(c-a)\Gamma(a)}\int_0^1t^{a-1}(1-t)^{c-a-1}(1-zt)^{-b}\,dt</math><br> | ||
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| + | <h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">초기하급수로 표현되는 함수의 예</h5> | ||
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| + | * [[타원적분]][[타원적분|]]<math>K(k) =\frac{\pi}{2}\,_2F_1(\frac{1}{2},\frac{1}{2};1;k^2)</math><br><math>E(k) =\frac{\pi}{2}\,_2F_1(\frac{1}{2},-\frac{1}{2};1;k^2)</math><br> | ||
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| 27번째 줄: | 39번째 줄: | ||
<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">타원적분과 초기하급수</h5> | <h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">타원적분과 초기하급수</h5> | ||
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| + | * <br>[[제1종타원적분 K (complete elliptic integral of the first kind)]]<br>[[제1종타원적분 K (complete elliptic integral of the first kind)|]]<math>K(k) = \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-k^2 \sin^2\theta}} = \int_0^{\frac{\pi}{2}}\sum_{n=0}^{\infty}\frac{(\frac{1}{2})_n}{n!} k^{2n}\sin^{2n}\theta{d\theta} </math><br> | ||
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<math>\int_0^{\frac{\pi}{2}}\sin^{2n}\theta{d\theta}=\frac{\pi}{2}\frac{(\frac{1}{2})_n}{(1)_n}</math> ([[#|감마함수]]) 이므로 | <math>\int_0^{\frac{\pi}{2}}\sin^{2n}\theta{d\theta}=\frac{\pi}{2}\frac{(\frac{1}{2})_n}{(1)_n}</math> ([[#|감마함수]]) 이므로 | ||
<math>K(k) = \frac{\pi}{2}\sum_{n=0}^{\infty}\frac{(\frac{1}{2})_n(\frac{1}{2})_n}{n!(1)_n}k^{2n} = \frac{\pi}{2}\,_2F_1(\frac{1}{2},\frac{1}{2};1;k^2)</math> | <math>K(k) = \frac{\pi}{2}\sum_{n=0}^{\infty}\frac{(\frac{1}{2})_n(\frac{1}{2})_n}{n!(1)_n}k^{2n} = \frac{\pi}{2}\,_2F_1(\frac{1}{2},\frac{1}{2};1;k^2)</math> | ||
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| 120번째 줄: | 126번째 줄: | ||
* [http://www.jstor.org/stable/2975319 On the Kummer Solutions of the Hypergeometric Equation]<br> | * [http://www.jstor.org/stable/2975319 On the Kummer Solutions of the Hypergeometric Equation]<br> | ||
** Reese T. Prosser, <cite style="line-height: 2em;">The American Mathematical Monthly</cite>, Vol. 101, No. 6 (Jun. - Jul., 1994), pp. 535-543 | ** Reese T. Prosser, <cite style="line-height: 2em;">The American Mathematical Monthly</cite>, Vol. 101, No. 6 (Jun. - Jul., 1994), pp. 535-543 | ||
| + | * Special values of the hypergeometric series | ||
| + | * Mathematical Proceedings of the Cambridge Philosophical Society (1991) volume: 109 issue: 2 page: 257 | ||
* [http://dx.doi.org/10.1007/BF01393999 Werte hypergeometrischer funktionen]<br> | * [http://dx.doi.org/10.1007/BF01393999 Werte hypergeometrischer funktionen]<br> | ||
** Jürgen Wolfart, Inventiones Mathematicae Volume 92, Number 1 / 1988년 2월 | ** Jürgen Wolfart, Inventiones Mathematicae Volume 92, Number 1 / 1988년 2월 | ||
2009년 12월 5일 (토) 15:09 판
이 항목의 스프링노트 원문주소
개요
- 정의
\(\,_2F_1(a,b;c;z)=\sum_{n=0}^{\infty} \frac{(a)_n(b)_n}{(c)_nn!}z^n, |z|<1\)
- 적분표현
\(\,_2F_1(a,b;c;z)=\frac{\Gamma(c)}{\Gamma(c-a)\Gamma(a)}\int_0^1t^{a-1}(1-t)^{c-a-1}(1-zt)^{-b}\,dt\)
초기하급수로 표현되는 함수의 예
- 타원적분[[타원적분|]]\(K(k) =\frac{\pi}{2}\,_2F_1(\frac{1}{2},\frac{1}{2};1;k^2)\)
\(E(k) =\frac{\pi}{2}\,_2F_1(\frac{1}{2},-\frac{1}{2};1;k^2)\)
피카드-Fuchs 미분방정식
- \(\,_2F_1(a,b;c;z)\) 는 다음 미분방정식의 해가 된다
\(z(1-z)\frac{d^2w}{dz^2}+(c-(a+b+1)z)\frac{dw}{dz}-abw = 0\) - 초기하 미분방정식(Hypergeometric differential equations) 참조
타원적분과 초기하급수
-
제1종타원적분 K (complete elliptic integral of the first kind)
[[제1종타원적분 K (complete elliptic integral of the first kind)|]]\(K(k) = \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-k^2 \sin^2\theta}} = \int_0^{\frac{\pi}{2}}\sum_{n=0}^{\infty}\frac{(\frac{1}{2})_n}{n!} k^{2n}\sin^{2n}\theta{d\theta} \)
(증명)
\(\int_0^{\frac{\pi}{2}}\sin^{2n}\theta{d\theta}=\frac{\pi}{2}\frac{(\frac{1}{2})_n}{(1)_n}\) (감마함수) 이므로
\(K(k) = \frac{\pi}{2}\sum_{n=0}^{\infty}\frac{(\frac{1}{2})_n(\frac{1}{2})_n}{n!(1)_n}k^{2n} = \frac{\pi}{2}\,_2F_1(\frac{1}{2},\frac{1}{2};1;k^2)\)
special values
\(\,_2F_1(a,b;c;1)=\dfrac{\Gamma(c)\,\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}\)
\(\frac{\pi}{2}\,_2F_1(\frac{1}{2},\frac{1}{2};1;\frac{1}{2})=K(\frac{1}{\sqrt{2}})=\frac{1}{4}B(1/4,1/4)=\frac{\Gamma(\frac{1}{4})^2}{4\sqrt{\pi}}=1.8540746773\cdots\)
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사전 형태의 자료
- http://ko.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/
- http://www.wolframalpha.com/input/?i=
- NIST Digital Library of Mathematical Functions
- The On-Line Encyclopedia of Integer Sequences
관련논문
- Transcendence of periods: the state of the art.
- M. Waldschmidt., Pure Appl. Math. Q. 2 (2006), 435-463.
- M. Waldschmidt., Pure Appl. Math. Q. 2 (2006), 435-463.
- On the Kummer Solutions of the Hypergeometric Equation
- Reese T. Prosser, The American Mathematical Monthly, Vol. 101, No. 6 (Jun. - Jul., 1994), pp. 535-543
- Special values of the hypergeometric series
- Mathematical Proceedings of the Cambridge Philosophical Society (1991) volume: 109 issue: 2 page: 257
- Werte hypergeometrischer funktionen
- Jürgen Wolfart, Inventiones Mathematicae Volume 92, Number 1 / 1988년 2월
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