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| − | <h5 style="line-height: 3.428em | + | <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%;">이 항목의 스프링노트 원문주소</h5> |
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| − | * 정의<br><math>\,_2F_1(a,b;c;z)=\sum_{n=0}^{\infty} \frac{(a)_n(b)_n}{(c)_nn!}z^n, |z|<1</math><br> | + | * 초기하급수로서의 정의<br><math>\,_2F_1(a,b;c;z)=\sum_{n=0}^{\infty} \frac{(a)_n(b)_n}{(c)_nn!}z^n, |z|<1</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> | * 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> | ||
| + | * 초기하급수를 해석적확<br> | ||
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| − | <h5 style=" | + | <h5 style="margin: 0px; line-height: 2em;">초기하급수로 표현되는 함수의 예</h5> |
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| − | <h5 style=" | + | <h5 style="margin: 0px; line-height: 2em;">피카드-Fuchs 미분방정식</h5> |
* <math>\,_2F_1(a,b;c;z)</math> 는 다음 미분방정식의 해가 된다<br><math>z(1-z)\frac{d^2w}{dz^2}+(c-(a+b+1)z)\frac{dw}{dz}-abw = 0</math><br> | * <math>\,_2F_1(a,b;c;z)</math> 는 다음 미분방정식의 해가 된다<br><math>z(1-z)\frac{d^2w}{dz^2}+(c-(a+b+1)z)\frac{dw}{dz}-abw = 0</math><br> | ||
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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> | * <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>\,_2F_1(a,b;c;1)=\dfrac{\Gamma(c)\,\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}</math> | <math>\,_2F_1(a,b;c;1)=\dfrac{\Gamma(c)\,\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}</math> | ||
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| − | <h5 style="line-height: 3.428em | + | <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%;">재미있는 사실</h5> |
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* [[수학사연표 (역사)|수학사연표]] | * [[수학사연표 (역사)|수학사연표]] | ||
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* [[periods]]<br> | * [[periods]]<br> | ||
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| − | <h5 style="line-height: 3.428em | + | <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%;">수학용어번역</h5> |
* http://www.google.com/dictionary?langpair=en|ko&q= | * http://www.google.com/dictionary?langpair=en|ko&q= | ||
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* http://ko.wikipedia.org/wiki/ | * http://ko.wikipedia.org/wiki/ | ||
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* http://www.wolframalpha.com/input/?i= | * http://www.wolframalpha.com/input/?i= | ||
* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions] | * [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions] | ||
| − | * [http://www.research.att.com/ | + | * [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]<br> |
** http://www.research.att.com/~njas/sequences/?q= | ** http://www.research.att.com/~njas/sequences/?q= | ||
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| − | <h5 style=" | + | <h5 style="margin: 0px; line-height: 2em;">expository articles</h5> |
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** 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 | ||
| − | * Ramanujan and hypergeometric and basic hypergeometric series | + | * [http://dx.doi.org/10.1070/RM1990v045n01ABEH002325 Ramanujan and hypergeometric and basic hypergeometric series]<br> |
| − | * R Askey 1990 Russ. Math. Surv. 45 37-86 | + | ** R Askey 1990 Russ. Math. Surv. 45 37-86 |
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| − | * [http://people.math.jussieu.fr/ | + | * [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> | ||
| − | * <br>[http://dx.doi.org/10.1016/S0022-314X | + | * <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> | ||
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* 도서내검색<br> | * 도서내검색<br> | ||
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* 네이버 뉴스 검색 (키워드 수정)<br> | * 네이버 뉴스 검색 (키워드 수정)<br> | ||
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* 구글 블로그 검색<br> | * 구글 블로그 검색<br> | ||
2009년 12월 7일 (월) 08:49 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
\(\,_2F_1(a,b;c;z)=\frac{\Gamma(c)}{\Gamma(c-a)\Gamma(a)}\int_0^1t^{a-1}(1-t)^{c-a-1}(1-zt)^{-b}\,dt\) - 초기하급수를 해석적확
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
\(\,_2F_1(a,b;c;1)=\dfrac{\Gamma(c)\,\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}\)
\(\frac{\pi}{2}\,_2F_1(\frac{1}{2},\frac{1}{2};1;\frac{1}{2})=K(\frac{1}{\sqrt{2}})=\frac{1}{4}B(1/4,1/4)=\frac{\Gamma(\frac{1}{4})^2}{4\sqrt{\pi}}=1.8540746773\cdots\)
재미있는 사실
역사
메모
관련된 항목들
수학용어번역
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
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
- 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
관련도서 및 추천도서
- 도서내검색
- 도서검색
관련기사
- 네이버 뉴스 검색 (키워드 수정)