"오일러-가우스 초기하함수2F1"의 두 판 사이의 차이
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<h5 style="margin: 0px; line-height: 2em;">contiguous 관계</h5> | <h5 style="margin: 0px; line-height: 2em;">contiguous 관계</h5> | ||
| − | * 파라미터 | + | * 두 초기하급수가 있을 때, 세 파라미터 중 두 개가 같고, 하나가 1만큼 다른 경우 contiguous라 함<br><math>_2F_1(a,b;c;z)</math>와 <math>_2F_1(a\pm1,b;c;z)</math><br><math>_2F_1(a,b;c;z)</math>와 <math>_2F_1(a1,b;c\pm1;z)</math><br> <br> |
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<h5 style="margin: 0px; line-height: 2em;">expository articles</h5> | <h5 style="margin: 0px; line-height: 2em;">expository articles</h5> | ||
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* [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> | ||
| 151번째 줄: | 151번째 줄: | ||
<h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">관련논문</h5> | <h5 style="margin: 0px; line-height: 3.428em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">관련논문</h5> | ||
| + | * [http://dx.doi.org/10.1016/j.cam.2005.05.016 On the contiguous relations of hypergeometric series]<br> | ||
| + | ** Medhat A. Rakha, Adel K. Ibrahim, Journal of Computational and Applied Mathematics, Volume 192, Issue 2, 1 August 2006, Pages 396-410 | ||
* [http://people.math.jussieu.fr/%7Emiw/articles/pdf/TranscendencePeriods.pdf Transcendence of periods: the state of the art.]<br> | * [http://people.math.jussieu.fr/%7Emiw/articles/pdf/TranscendencePeriods.pdf Transcendence of periods: the state of the art.]<br> | ||
** M. Waldschmidt., Pure Appl. Math. Q. 2 (2006), 435-463.<br> | ** M. Waldschmidt., Pure Appl. Math. Q. 2 (2006), 435-463.<br> | ||
| − | * | + | * [http://dx.doi.org/10.1016/S0022-314X%2803%2900042-8 Exceptional sets of hypergeometric series]<br> |
** Natália Archinard, Journal of Number Theory Volume 101, Issue 2, August 2003, Pages 244-269<br> | ** Natália Archinard, Journal of Number Theory Volume 101, Issue 2, August 2003, Pages 244-269<br> | ||
2009년 12월 7일 (월) 08:25 판
이 항목의 스프링노트 원문주소
개요
- 초기하급수로서의 정의
\(\,_2F_1(a,b;c;z)=\sum_{n=0}^{\infty} \frac{(a)_n(b)_n}{(c)_nn!}z^n, |z|<1\)
- 적분표현
\(\,_2F_1(a,b;c;z)=\frac{\Gamma(c)}{\Gamma(c-a)\Gamma(a)}\int_0^1t^{a-1}(1-t)^{c-a-1}(1-zt)^{-b}\,dt\) - 초기하급수의 해석적확장을 통해 얻어진 함수를 초기하함수라 함
초기하급수로 표현되는 함수의 예
- 타원적분
\(K(k) =\frac{\pi}{2}\,_2F_1(\frac{1}{2},\frac{1}{2};1;k^2)\)
\(E(k) =\frac{\pi}{2}\,_2F_1(\frac{1}{2},-\frac{1}{2};1;k^2)\)
오일러의 항등식
\(_2F_1 (a,b;c;z) = (1-z)^{c-a-b}{}_2F_1 (c-a, c-b;c ; z)\)
\(_2F_1 (a,b;c;z) = (1-z)^{-a} {}_2F_1 (a, c-b;c ; \frac{z}{z-1})\)
contiguous 관계
- 두 초기하급수가 있을 때, 세 파라미터 중 두 개가 같고, 하나가 1만큼 다른 경우 contiguous라 함
\(_2F_1(a,b;c;z)\)와 \(_2F_1(a\pm1,b;c;z)\)
\(_2F_1(a,b;c;z)\)와 \(_2F_1(a1,b;c\pm1;z)\)
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
\(\,_2F_1(a,b;c;1)=\dfrac{\Gamma(c)\,\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}\)
\(\frac{\pi}{2}\,_2F_1(\frac{1}{2},\frac{1}{2};1;\frac{1}{2})=K(\frac{1}{\sqrt{2}})=\frac{1}{4}B(1/4,1/4)=\frac{\Gamma(\frac{1}{4})^2}{4\sqrt{\pi}}=1.8540746773\cdots\)
재미있는 사실
역사
메모
관련된 항목들
수학용어번역
사전 형태의 자료
- http://ko.wikipedia.org/wiki/
- [1]http://en.wikipedia.org/wiki/hypergeometric_functions
- http://en.wikipedia.org/wiki/List_of_hypergeometric_identities
- http://en.wikipedia.org/wiki/
- http://www.wolframalpha.com/input/?i=
- NIST Digital Library of Mathematical Functions
- The On-Line Encyclopedia of Integer Sequences
expository articles
- 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
- Ramanujan and hypergeometric and basic hypergeometric series
- R Askey 1990 Russ. Math. Surv. 45 37-86
관련논문
- On the contiguous relations of hypergeometric series
- Medhat A. Rakha, Adel K. Ibrahim, Journal of Computational and Applied Mathematics, Volume 192, Issue 2, 1 August 2006, Pages 396-410
- 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.
- Exceptional sets of hypergeometric series
- Natália Archinard, Journal of Number Theory Volume 101, Issue 2, August 2003, Pages 244-269
- Natália Archinard, Journal of Number Theory Volume 101, Issue 2, August 2003, Pages 244-269
- Special values of the hypergeometric series III
- Joyce, G. S.; Zucker, I. J., Mathematical Proceedings of the Cambridge Philosophical Society (2002), 133 : 213-222
- Special values of the hypergeometric series II
- Joyce, G. S.; Zucker, I. J., Mathematical Proceedings of the Cambridge Philosophical Society (2001), 131 : 309-319
- Special values of the hypergeometric series
- Joyce, G. S.; Zucker, I. J., 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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