"오일러-가우스 초기하함수2F1"의 두 판 사이의 차이
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| 1번째 줄: | 1번째 줄: | ||
==개요== | ==개요== | ||
| − | * 초기하급수:<math>\,_2F_1(a,b;c;z)=\sum_{n=0}^{\infty} \frac{(a)_n(b)_n}{(c)_nn!}z^n, |z|<1</math> | + | * 초기하급수:<math>\,_2F_1(a,b;c;z)=\sum_{n=0}^{\infty} \frac{(a)_n(b)_n}{(c)_nn!}z^n, |z|<1</math> |
| − | * 적분표현:<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 | + | 여기서 <math>(a)_n=a(a+1)(a+2)...(a+n-1)</math>에 대해서는 [[포흐하머 (Pochhammer) 기호]] 항목 참조 |
| − | * 초기하급수의 해석적확장을 통해 얻어진 함수를 초기하함수라 함 | + | * 적분표현:<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> |
| − | * 오일러, 가우스, 쿰머, 리만,슈워츠 등의 연구 | + | * 초기하급수의 해석적확장을 통해 얻어진 함수를 초기하함수라 함 |
| + | * 오일러, 가우스, 쿰머, 리만,슈워츠 등의 연구 | ||
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==초기하급수로 표현되는 함수의 예== | ==초기하급수로 표현되는 함수의 예== | ||
* 많은 special function 은 초기하함수의 파라메터를 변화시켜 얻어짐 | * 많은 special function 은 초기하함수의 파라메터를 변화시켜 얻어짐 | ||
| − | * [[제1종타원적분 K (complete elliptic integral of the first kind)]]:<math>K(k) =\frac{\pi}{2}\,_2F_1(\frac{1}{2},\frac{1}{2};1;k^2)</math | + | * [[제1종타원적분 K (complete elliptic integral of the first kind)]]:<math>K(k) =\frac{\pi}{2}\,_2F_1(\frac{1}{2},\frac{1}{2};1;k^2)</math> |
| − | * [[제2종타원적분 E (complete elliptic integral of the second kind)]]:<math>E(k) =\frac{\pi}{2}\,_2F_1(\frac{1}{2},-\frac{1}{2};1;k^2)</math | + | * [[제2종타원적분 E (complete elliptic integral of the second kind)]]:<math>E(k) =\frac{\pi}{2}\,_2F_1(\frac{1}{2},-\frac{1}{2};1;k^2)</math> |
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==초기하 미분방정식== | ==초기하 미분방정식== | ||
| − | * <math>w(z)=\,_2F_1(a,b;c;z)</math> | + | * <math>w(z)=\,_2F_1(a,b;c;z)</math> 는 다음 피카드-Fuchs 형태의 미분방정식의 해가 된다 |
| − | * 이 미분방정식을 [[초기하 미분방정식(Hypergeometric differential equations)]] 이라 부른다 | + | :<math>z(1-z)\frac{d^2w}{dz^2}+(c-(a+b+1)z)\frac{dw}{dz}-abw = 0</math> |
| + | * 이 미분방정식을 [[초기하 미분방정식(Hypergeometric differential equations)]] 이라 부른다 | ||
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==오일러의 변환 공식== | ==오일러의 변환 공식== | ||
| 39번째 줄: | 41번째 줄: | ||
<math>_2F_1 (a,b;c;z) = (1-z)^{c-a-b}{}_2F_1 (c-a, c-b;c ; z)</math> | <math>_2F_1 (a,b;c;z) = (1-z)^{c-a-b}{}_2F_1 (c-a, c-b;c ; z)</math> | ||
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| + | ;증명 | ||
다음 적분표현을 활용 | 다음 적분표현을 활용 | ||
<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> | <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> | ||
| − | 위의 | + | 위의 우변에서 <math>t\to 1-t</math>, <math>t\to \frac{t}{1-z-tz}</math>, <math>t\to \frac{1-t}{1-tz}</math>의 변환을 이용하면 항등식이 얻어진다. ■ |
| − | * http://mathworld.wolfram.com/EulersHypergeometricTransformations.html | + | * http://mathworld.wolfram.com/EulersHypergeometricTransformations.html |
| − | * 쿰머의 [[초기하 미분방정식(Hypergeometric differential equations)]]에 대한 24개의 해를 표현하는데 사용됨 | + | * 쿰머의 [[초기하 미분방정식(Hypergeometric differential equations)]]에 대한 24개의 해를 표현하는데 사용됨 |
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==contiguous 관계== | ==contiguous 관계== | ||
| − | * [[초기하함수 2F1의 contiguous 관계]] | + | * [[초기하함수 2F1의 contiguous 관계]] |
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==타원적분과 초기하급수== | ==타원적분과 초기하급수== | ||
| − | * [[제1종타원적분 K (complete elliptic integral of the first kind)]]:<math>K(k) =\frac{\pi}{2}\,_2F_1(\frac{1}{2},\frac{1}{2};1;k^2)</math | + | * [[제1종타원적분 K (complete elliptic integral of the first kind)]]:<math>K(k) =\frac{\pi}{2}\,_2F_1(\frac{1}{2},\frac{1}{2};1;k^2)</math> |
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==모듈라 함수와의 관계== | ==모듈라 함수와의 관계== | ||
| − | * [[라마누잔과 파이]] | + | * [[라마누잔과 파이]] |
| − | * '''[BB1998]'''[http://www.amazon.com/PI-AGM-Analytic-Computational-Complexity/dp/047131515X Pi and the AGM] | + | * '''[BB1998]'''[http://www.amazon.com/PI-AGM-Analytic-Computational-Complexity/dp/047131515X Pi and the AGM] |
| − | * Jonathan M. Borwein, Peter B. Borwein, | + | * Jonathan M. Borwein, Peter B. Borwein, Wiley-Interscience (July 13, 1998) 179,180p |
| − | * '''[Nes2002] 159p''' | + | * '''[Nes2002] 159p''' |
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==슈워츠 s-함수== | ==슈워츠 s-함수== | ||
| − | * [[슈바르츠 삼각형 함수|슈워츠 s-함수]] | + | * [[슈바르츠 삼각형 함수|슈워츠 s-함수]] |
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==special values== | ==special values== | ||
| − | * Chu-Vandermonde 공식:<math>\,_2F_1(-n,b;c;1)=\dfrac{(c-b)_{n}}{(c)_{n}}</math | + | * Chu-Vandermonde 공식:<math>\,_2F_1(-n,b;c;1)=\dfrac{(c-b)_{n}}{(c)_{n}}</math> 아래 가우스 공식에서 <math>a=-n</math>인 경우에 얻어진다 |
| − | * 가우스 공식:<math>\,_2F_1(a,b;c;1)=\dfrac{\Gamma(c)\,\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}</math | + | * 가우스 공식:<math>\,_2F_1(a,b;c;1)=\dfrac{\Gamma(c)\,\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}</math> |
| − | * 위의 두 식에 | + | * 위의 두 식에 대해서는 [[초기하급수의 합공식|초기하 급수의 합공식]] |
| − | * [[렘니스케이트(lemniscate) 곡선의 길이와 타원적분|렘니스케이트(lemniscate) 곡선과 타원적분]]:<math>\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</math | + | * [[렘니스케이트(lemniscate) 곡선의 길이와 타원적분|렘니스케이트(lemniscate) 곡선과 타원적분]]:<math>\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</math> |
| − | * http://mathworld.wolfram.com/HypergeometricFunction.html:<math>_2F_1(\frac{1}{3},\frac{2}{3};\frac{5}{6};\frac{27}{32})=\frac{8}{5}</math>:<math>_2F_1(\frac{1}{4},\frac{1}{2};\frac{3}{4};\frac{80}{81})=\frac{9}{5}</math>:<math>_2F_1(\frac{1}{8},\frac{3}{8};\frac{1}{2};\frac{2400}{2401})=\frac{2}{3}\sqrt{7}</math>:<math>_2F_1(\frac{1}{6},\frac{1}{3};\frac{1}{2};\frac{25}{27})=\frac{3}{4}\sqrt{3}</math>:<math>_2F_1(\frac{1}{6},\frac{1}{2};\frac{2}{3};\frac{125}{128})=\frac{4}{3}\sqrt[6]2</math>:<math>_2F_1(\frac{1}{12},\frac{5}{12};\frac{1}{2};\frac{1323}{1331})=\frac{3}{4}\sqrt[4]{11}</math>:<math>_2F_1(\frac{1}{12},\frac{5}{12};\frac{1}{2};\frac{121}{125})=\frac{\sqrt[6]{2}\sqrt[4]{15}}{4\sqrt{\pi}}\frac{\Gamma(\frac{1}{3})^3}{\Gamma(\frac{1}{4})^2}(1+\sqrt{3})</math | + | * http://mathworld.wolfram.com/HypergeometricFunction.html:<math>_2F_1(\frac{1}{3},\frac{2}{3};\frac{5}{6};\frac{27}{32})=\frac{8}{5}</math>:<math>_2F_1(\frac{1}{4},\frac{1}{2};\frac{3}{4};\frac{80}{81})=\frac{9}{5}</math>:<math>_2F_1(\frac{1}{8},\frac{3}{8};\frac{1}{2};\frac{2400}{2401})=\frac{2}{3}\sqrt{7}</math>:<math>_2F_1(\frac{1}{6},\frac{1}{3};\frac{1}{2};\frac{25}{27})=\frac{3}{4}\sqrt{3}</math>:<math>_2F_1(\frac{1}{6},\frac{1}{2};\frac{2}{3};\frac{125}{128})=\frac{4}{3}\sqrt[6]2</math>:<math>_2F_1(\frac{1}{12},\frac{5}{12};\frac{1}{2};\frac{1323}{1331})=\frac{3}{4}\sqrt[4]{11}</math>:<math>_2F_1(\frac{1}{12},\frac{5}{12};\frac{1}{2};\frac{121}{125})=\frac{\sqrt[6]{2}\sqrt[4]{15}}{4\sqrt{\pi}}\frac{\Gamma(\frac{1}{3})^3}{\Gamma(\frac{1}{4})^2}(1+\sqrt{3})</math> |
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==역사== | ==역사== | ||
| 108번째 줄: | 109번째 줄: | ||
* [[수학사 연표]] | * [[수학사 연표]] | ||
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==메모== | ==메모== | ||
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==관련된 항목들== | ==관련된 항목들== | ||
| − | * [[periods]] | + | * [[periods]] |
| − | * [[무리수와 초월수]] | + | * [[무리수와 초월수]] |
| − | * [[오일러 베타적분(베타함수)|오일러 베타적분]] | + | * [[오일러 베타적분(베타함수)|오일러 베타적분]] |
| − | * [[직교다항식과 special functions|Special functions]] | + | * [[직교다항식과 special functions|Special functions]] |
| − | * [[맴돌이군과 미분방정식]] | + | * [[맴돌이군과 미분방정식]] |
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==매스매티카 파일 및 계산 리소스== | ==매스매티카 파일 및 계산 리소스== | ||
| 139번째 줄: | 140번째 줄: | ||
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| − | ==사전 | + | ==사전 형태의 자료== |
* [http://ko.wikipedia.org/wiki/%EC%B4%88%EA%B8%B0%ED%95%98%ED%95%A8%EC%88%98 http://ko.wikipedia.org/wiki/초기하함수] | * [http://ko.wikipedia.org/wiki/%EC%B4%88%EA%B8%B0%ED%95%98%ED%95%A8%EC%88%98 http://ko.wikipedia.org/wiki/초기하함수] | ||
| 149번째 줄: | 150번째 줄: | ||
* http://en.wikipedia.org/wiki/Frobenius_solution_to_the_hypergeometric_equation | * http://en.wikipedia.org/wiki/Frobenius_solution_to_the_hypergeometric_equation | ||
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==리뷰논문, 에세이, 강의노트== | ==리뷰논문, 에세이, 강의노트== | ||
| − | * '''[Nes2002]'''[http://books.google.com/books?id=Up-XxkiTtdsC&pg=PA148&lpg=PA148&dq=On+the+Algebraic+Independence+of+Numbers+Yu.V.+Nesterenko&source=bl&ots=yOVhiH5ukL&sig=x0GqVIluMqw-_Iaf3tXtKxam50Q&hl=ko&ei=KIwRTPiwB4rcNcSE8ccF&sa=X&oi=book_result&ct=result&resnum=3&ved=0CCQQ6AEwAg#v=onepage&q=On%20the%20Algebraic%20Independence%20of%20Numbers%20Yu.V.%20Nesterenko&f=false On the Algebraic Independence of Numbers] | + | * '''[Nes2002]'''[http://books.google.com/books?id=Up-XxkiTtdsC&pg=PA148&lpg=PA148&dq=On+the+Algebraic+Independence+of+Numbers+Yu.V.+Nesterenko&source=bl&ots=yOVhiH5ukL&sig=x0GqVIluMqw-_Iaf3tXtKxam50Q&hl=ko&ei=KIwRTPiwB4rcNcSE8ccF&sa=X&oi=book_result&ct=result&resnum=3&ved=0CCQQ6AEwAg#v=onepage&q=On%20the%20Algebraic%20Independence%20of%20Numbers%20Yu.V.%20Nesterenko&f=false On the Algebraic Independence of Numbers] |
| − | ** Yu.V. Nesterenko, in <em style="">A panorama in number theory, or, The view from Baker's garden</em> (by Alan Baker,Gisbert Wüstholz), 2002 | + | ** Yu.V. Nesterenko, in <em style="">A panorama in number theory, or, The view from Baker's garden</em> (by Alan Baker,Gisbert Wüstholz), 2002 |
| − | * [http://www.jstor.org/stable/2975319 On the Kummer Solutions of the Hypergeometric Equation] | + | * [http://www.jstor.org/stable/2975319 On the Kummer Solutions of the Hypergeometric Equation] |
| − | ** Reese T. Prosser, | + | ** Reese T. Prosser, <cite style="line-height: 2em;">The American Mathematical Monthly</cite>, Vol. 101, No. 6 (Jun. - Jul., 1994), pp. 535-543 |
| − | * [http://dx.doi.org/10.1070/RM1990v045n01ABEH002325 Ramanujan and hypergeometric and basic hypergeometric series] | + | * [http://dx.doi.org/10.1070/RM1990v045n01ABEH002325 Ramanujan and hypergeometric and basic hypergeometric series] |
** R Askey 1990 Russ. Math. Surv. 45 37-86 | ** R Askey 1990 Russ. Math. Surv. 45 37-86 | ||
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==관련논문== | ==관련논문== | ||
| − | * [http://dx.doi.org/10.1016/j.cam.2005.05.016 On the contiguous relations of hypergeometric series] | + | * [http://dx.doi.org/10.1016/j.cam.2005.05.016 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 | ** 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.] | + | * [http://people.math.jussieu.fr/%7Emiw/articles/pdf/TranscendencePeriods.pdf Transcendence of periods: the state of the art.] |
| − | ** M. Waldschmidt., | + | ** M. Waldschmidt., Pure Appl. Math. Q. 2 (2006), 435-463. |
| − | * [http://dx.doi.org/10.1016/S0022-314X%2803%2900042-8 Exceptional sets of hypergeometric series] | + | * [http://dx.doi.org/10.1016/S0022-314X%2803%2900042-8 Exceptional sets of hypergeometric series] |
| − | ** Natália Archinard, | + | ** Natália Archinard, Journal of Number Theory Volume 101, Issue 2, August 2003, Pages 244-269 |
* Thorsley, Michael D., and Marita C. Chidichimo. 2001. “An Asymptotic Expansion for the Hypergeometric Function 2F1(a,b;c;x).” Journal of Mathematical Physics 42 (4) (April 1): 1921–1930. doi:doi:10.1063/1.1353185. http://jmp.aip.org/resource/1/jmapaq/v42/i4/p1921_s1 | * Thorsley, Michael D., and Marita C. Chidichimo. 2001. “An Asymptotic Expansion for the Hypergeometric Function 2F1(a,b;c;x).” Journal of Mathematical Physics 42 (4) (April 1): 1921–1930. doi:doi:10.1063/1.1353185. http://jmp.aip.org/resource/1/jmapaq/v42/i4/p1921_s1 | ||
| − | * [http://dx.doi.org/10.1017/S0305004102005923 Special values of the hypergeometric series III] | + | * [http://dx.doi.org/10.1017/S0305004102005923 Special values of the hypergeometric series III] |
| − | ** Joyce, G. S.; Zucker, I. J., | + | ** Joyce, G. S.; Zucker, I. J., Mathematical Proceedings of the Cambridge Philosophical Society (2002), 133 : 213-222 |
| − | * [http://dx.doi.org/10.1017/S0305004101005254 Special values of the hypergeometric series II] | + | * [http://dx.doi.org/10.1017/S0305004101005254 Special values of the hypergeometric series II] |
| − | ** Joyce, G. S.; Zucker, I. J., | + | ** Joyce, G. S.; Zucker, I. J., Mathematical Proceedings of the Cambridge Philosophical Society (2001), 131 : 309-319 |
| − | * Special values of the hypergeometric series | + | * Special values of the hypergeometric series |
| − | ** Joyce, G. S.; Zucker, I. J., | + | ** Joyce, G. S.; Zucker, I. J., Mathematical Proceedings of the Cambridge Philosophical Society (1991) volume: 109 issue: 2 page: 257 |
| − | * [http://dx.doi.org/10.1007/BF01393999 Werte hypergeometrischer funktionen] | + | * [http://dx.doi.org/10.1007/BF01393999 Werte hypergeometrischer funktionen] |
| − | ** Jürgen Wolfart, | + | ** Jürgen Wolfart, Inventiones Mathematicae Volume 92, Number 1 / 1988년 2월 |
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[[분류:리만곡면론]] | [[분류:리만곡면론]] | ||
2014년 1월 3일 (금) 16:03 판
개요
- 초기하급수\[\,_2F_1(a,b;c;z)=\sum_{n=0}^{\infty} \frac{(a)_n(b)_n}{(c)_nn!}z^n, |z|<1\]
여기서 \((a)_n=a(a+1)(a+2)...(a+n-1)\)에 대해서는 포흐하머 (Pochhammer) 기호 항목 참조
- 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
- 오일러, 가우스, 쿰머, 리만,슈워츠 등의 연구
초기하급수로 표현되는 함수의 예
- 많은 special function 은 초기하함수의 파라메터를 변화시켜 얻어짐
- 제1종타원적분 K (complete elliptic integral of the first kind)\[K(k) =\frac{\pi}{2}\,_2F_1(\frac{1}{2},\frac{1}{2};1;k^2)\]
- 제2종타원적분 E (complete elliptic integral of the second kind)\[E(k) =\frac{\pi}{2}\,_2F_1(\frac{1}{2},-\frac{1}{2};1;k^2)\]
초기하 미분방정식
- \(w(z)=\,_2F_1(a,b;c;z)\) 는 다음 피카드-Fuchs 형태의 미분방정식의 해가 된다
\[z(1-z)\frac{d^2w}{dz^2}+(c-(a+b+1)z)\frac{dw}{dz}-abw = 0\]
- 이 미분방정식을 초기하 미분방정식(Hypergeometric differential equations) 이라 부른다
오일러의 변환 공식
\(_2F_1 (a,b;c;z) = (1-z)^{-a} {}_2F_1 (a, c-b;c ; \frac{z}{z-1})\)
\(_2F_1 (a,b;c;z) = (1-z)^{-b}{}_2F_1(c-a,b;c;\frac{z}{z-1})\)
\(_2F_1 (a,b;c;z) = (1-z)^{c-a-b}{}_2F_1 (c-a, c-b;c ; z)\)
- 증명
다음 적분표현을 활용
\(\,_2F_1(a,b;c;z)=\frac{\Gamma(c)}{\Gamma(c-a)\Gamma(a)}\int_0^1t^{a-1}(1-t)^{c-a-1}(1-zt)^{-b}\,dt\)
위의 우변에서 \(t\to 1-t\), \(t\to \frac{t}{1-z-tz}\), \(t\to \frac{1-t}{1-tz}\)의 변환을 이용하면 항등식이 얻어진다. ■
- http://mathworld.wolfram.com/EulersHypergeometricTransformations.html
- 쿰머의 초기하 미분방정식(Hypergeometric differential equations)에 대한 24개의 해를 표현하는데 사용됨
contiguous 관계
타원적분과 초기하급수
- 제1종타원적분 K (complete elliptic integral of the first kind)\[K(k) =\frac{\pi}{2}\,_2F_1(\frac{1}{2},\frac{1}{2};1;k^2)\]
모듈라 함수와의 관계
- 라마누잔과 파이
- [BB1998]Pi and the AGM
- Jonathan M. Borwein, Peter B. Borwein, Wiley-Interscience (July 13, 1998) 179,180p
- [Nes2002] 159p
슈워츠 s-함수
special values
- Chu-Vandermonde 공식\[\,_2F_1(-n,b;c;1)=\dfrac{(c-b)_{n}}{(c)_{n}}\] 아래 가우스 공식에서 \(a=-n\)인 경우에 얻어진다
- 가우스 공식\[\,_2F_1(a,b;c;1)=\dfrac{\Gamma(c)\,\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}\]
- 위의 두 식에 대해서는 초기하 급수의 합공식
- 렘니스케이트(lemniscate) 곡선과 타원적분\[\frac{\pi}{2}\,_2F_1(\frac{1}{2},\frac{1}{2};1;\frac{1}{2})=K(\frac{1}{\sqrt{2}})=\frac{1}{4}B(1/4,1/4)=\frac{\Gamma(\frac{1}{4})^2}{4\sqrt{\pi}}=1.8540746773\cdots\]
- http://mathworld.wolfram.com/HypergeometricFunction.html\[_2F_1(\frac{1}{3},\frac{2}{3};\frac{5}{6};\frac{27}{32})=\frac{8}{5}\]\[_2F_1(\frac{1}{4},\frac{1}{2};\frac{3}{4};\frac{80}{81})=\frac{9}{5}\]\[_2F_1(\frac{1}{8},\frac{3}{8};\frac{1}{2};\frac{2400}{2401})=\frac{2}{3}\sqrt{7}\]\[_2F_1(\frac{1}{6},\frac{1}{3};\frac{1}{2};\frac{25}{27})=\frac{3}{4}\sqrt{3}\]\[_2F_1(\frac{1}{6},\frac{1}{2};\frac{2}{3};\frac{125}{128})=\frac{4}{3}\sqrt[6]2\]\[_2F_1(\frac{1}{12},\frac{5}{12};\frac{1}{2};\frac{1323}{1331})=\frac{3}{4}\sqrt[4]{11}\]\[_2F_1(\frac{1}{12},\frac{5}{12};\frac{1}{2};\frac{121}{125})=\frac{\sqrt[6]{2}\sqrt[4]{15}}{4\sqrt{\pi}}\frac{\Gamma(\frac{1}{3})^3}{\Gamma(\frac{1}{4})^2}(1+\sqrt{3})\]
역사
메모
관련된 항목들
매스매티카 파일 및 계산 리소스
사전 형태의 자료
- http://ko.wikipedia.org/wiki/초기하함수
- http://en.wikipedia.org/wiki/hypergeometric_functions
- http://en.wikipedia.org/wiki/List_of_hypergeometric_identities
- http://en.wikipedia.org/wiki/hypergeometric_differential_equation
- http://en.wikipedia.org/wiki/Frobenius_solution_to_the_hypergeometric_equation
리뷰논문, 에세이, 강의노트
- [Nes2002]On the Algebraic Independence of Numbers
- Yu.V. Nesterenko, in A panorama in number theory, or, The view from Baker's garden (by Alan Baker,Gisbert Wüstholz), 2002
- 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.
- Exceptional sets of hypergeometric series
- Natália Archinard, Journal of Number Theory Volume 101, Issue 2, August 2003, Pages 244-269
- Thorsley, Michael D., and Marita C. Chidichimo. 2001. “An Asymptotic Expansion for the Hypergeometric Function 2F1(a,b;c;x).” Journal of Mathematical Physics 42 (4) (April 1): 1921–1930. doi:doi:10.1063/1.1353185. http://jmp.aip.org/resource/1/jmapaq/v42/i4/p1921_s1
- 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