"라마누잔과 파이"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) |
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| (사용자 2명의 중간 판 45개는 보이지 않습니다) | |||
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| − | + | ==개요== | |
| − | * 라마누잔은 1914년에 다음과 같은 공식을 | + | * 라마누잔은 1914년에 다음과 같은 공식을 발표 '''[RAM1914]''':<math>\frac{1}{\pi}= \frac{2\sqrt2}{9801}\sum_{n=0}^{\infty}\frac{(4n)!(1103+26390n)}{(n!)^{4}396^{4n}}</math> |
| − | + | * Chudnovsky 형제 '''[CHU88]''' | |
| − | + | :<math>\frac{426880 \sqrt{10005}}{\pi} = \sum_{k=0}^\infty \frac{(6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 (-640320)^{3k}}\!</math> | |
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| − | + | * [[숫자 163]] | |
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| + | ==정의와 미리 알아야 할 것들== | ||
| − | + | * [[자코비 세타함수]], [[라마누잔의 class invariants]], [[타원적분]] 참조 | |
| − | + | <math>q=e^{2\pi i \tau}</math> | |
| − | < | + | <math>\theta_{2}(\tau)= \sum_{n=-\infty}^\infty q^{(n+\frac{1}{2})^2/2}</math> |
| − | + | <math>\theta_3(\tau)=\sum_{n=-\infty}^\infty q^{n^2/2}</math> | |
<math>\theta_{4}(\tau)= \sum_{n=-\infty}^\infty (-1)^n q^{n^2/2}</math> | <math>\theta_{4}(\tau)= \sum_{n=-\infty}^\infty (-1)^n q^{n^2/2}</math> | ||
| 25번째 줄: | 27번째 줄: | ||
<math>K(k) = \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-k^2 \sin^2\theta}}</math> | <math>K(k) = \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-k^2 \sin^2\theta}}</math> | ||
| − | <math>E(k) = \int_0^{\frac{\pi}{2}} \sqrt{1-k^2 \sin^2\theta | + | <math>E(k) = \int_0^{\frac{\pi}{2}} \sqrt{1-k^2 \sin^2\theta}d\theta</math> |
<math>k'=\sqrt{1-k^2}=\frac{\theta_4^2(\tau)}{\theta_3^2(\tau)}</math> | <math>k'=\sqrt{1-k^2}=\frac{\theta_4^2(\tau)}{\theta_3^2(\tau)}</math> | ||
| 33번째 줄: | 35번째 줄: | ||
<math>E'(k) = E(k')</math> | <math>E'(k) = E(k')</math> | ||
| − | * 위의 함수들을 이용하여, | + | * 위의 함수들을 이용하여, 양수 <math>r</math>에 대하여 다음을 정의 |
<math>\lambda^{*}(r):=k(i\sqrt{r})</math> | <math>\lambda^{*}(r):=k(i\sqrt{r})</math> | ||
| 39번째 줄: | 41번째 줄: | ||
<math>\alpha(r):=\frac{E'}{K}-\frac{\pi}{4K^2}</math> | <math>\alpha(r):=\frac{E'}{K}-\frac{\pi}{4K^2}</math> | ||
| − | + | * [[라마누잔의 class invariants]]:<math>g_n:=2^{-1/4}\frac{\eta(\frac{\sqrt{-n}}{2})}{\eta(\sqrt{-n})}</math> | |
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| − | + | ==singular value function == | |
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| − | + | * 타원적분이 만족시키는 르장드르 항등식 ([[산술기하평균함수(AGM)와 파이값의 계산|AGM과 파이값의 계산]]) | |
| + | :<math>E(k)K'(k)+E'(k)K(k)-K(k)K'(k)=\frac{\pi}{2}\label{leg}</math> | ||
| + | * 타원적분의 성질 | ||
| + | :<math>K'(\lambda^{*}(r))=\sqrt{r}K(\lambda^{*}(r))\label{ell}</math> | ||
| + | * \ref{leg}와 \ref{ell}로부터 다음을 얻는다 | ||
| + | :<math>\alpha(r)=\frac{\pi}{4K^2}-\sqrt{r}(\frac{E}{K}-1)</math> | ||
| + | * 여기에 타원적분이 만족시키는 미분방정식 | ||
| + | :<math>\frac{dK}{dk}=\frac{E-k'^2K}{kk'^2}</math> 을 사용하면 | ||
| + | :<math>\alpha(r)=\frac{1}{\pi}(\frac{\pi}{2K})^2-\sqrt{r}(kk'^2\frac{\dot{K}}{K}-k^2)</math> 를 얻게 되고, 이를 다시 쓰면 | ||
| + | :<math>\frac{1}{\pi}=\sqrt{N}k_Nk'^2_N\frac{4K\dot{K}}{\pi^2}+[\alpha(N)-\sqrt{N}k^2_N]\frac{4K^2}{\pi^2}</math> | ||
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| − | < | + | * <math>[\frac{2}{\pi}K(k)]^2 =m(k)F(y(k))</math> 꼴로 쓰여질때, 양변을 미분하면 다음을 얻는다:<math>\frac{4K\dot{K}}{\pi^2}=\frac{1}{2}\dot{m}F+\frac{1}{2}m\dot{y}\dot{F}(y)</math> |
| + | * 초기하급수를 다음과 같이 쓰면 | ||
| + | :<math>F(y)=\sum_{n=0}^{\infty}a_ny^n</math> | ||
| + | 다음을 얻는다 | ||
| + | :<math>\frac{1}{\pi}=\sum_{n=0}^\infty a_n[\frac{\sqrt{N}}{2}k{k'}^2\dot{m}+[\alpha(N)-\sqrt{N}k^2_N]m+\frac{n\sqrt{N}}{2}m\frac{\dot{y}}{y}kk'^2]y^n</math> | ||
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| − | + | ==라마누잔 파이 공식의 유도== | |
| − | + | * 아래의 prop, thm 번호는 '''[BB1998] '''참조 | |
| + | * [[초기하급수(Hypergeometric series)]] 항목의 Clausen 항등식이 중요하게 사용됨 | ||
| + | * prop 5.6:<math>\frac{2}{\pi}K_s(h) = \,_2F_1(\frac{1}{4}-\frac{s}{2},\frac{1}{4}+\frac{s}{2};1;(2hh')^2)</math>:<math>[\frac{2}{\pi}K_s(h)]^2 = \,_2F_1(\frac{1}{2}-s,\frac{1}{2}+s,\frac{1}{2};1,1;(2hh')^2)</math> | ||
| + | * prop 5.7:<math>K_{1/4}(h)=(1+k^2)^{1/2}K(k)</math> if <math>2hh'=[\frac{g^{12}+g^{-12}}{2}]^{-1}</math> | ||
| + | * Thm 5.6:<math>\frac{2}{\pi}K(k) =(1+k^2)^{-1/2} \,_2F_1(\frac{1}{8},\frac{3}{8};1;[\frac{g^{12}+g^{-12}}{2}]^{-2})</math> | ||
| + | * Thm 5.7:<math>[\frac{2}{\pi}K(k)]^2 =(1+k^2)^{-1} \,_3F_2(\frac{1}{4},\frac{3}{4},\frac{1}{2};1,1;[\frac{g^{12}+g^{-12}}{2}]^{-2})</math> | ||
| + | * (5.5.16):<math>\frac{1}{\pi}=\sum_{n=0}^{\infty}\frac{(\frac{1}{4})_n(\frac{1}{2})_n(\frac{3}{4})_n}{(n!)^3}d_n(N)x_N^{2n+1}</math>:<math>x_N=(\frac{g_N^{12}+g_N^{-12}}{2})^{-1}</math>:<math>d_n(N)=[\frac{\alpha(N)x_N^{-1}}{1+k_N^2}-\frac{\sqrt{N}}{4}g_N^{-12}]+n\sqrt N(\frac{g_N^{12}-g_N^{-12}}{2})</math> | ||
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| − | < | + | * <math>N=58</math> 일 때 |
| + | :<math>x_{58}=\frac{1}{99^2}=\frac{1}{9801},</math> | ||
| + | :<math>d_n(58)=(1103+26390n)2\sqrt 2</math> 에서 다음을 얻는다 | ||
| + | :<math>\frac{1}{\pi}= \frac{2\sqrt2}{9801}\sum_{n=0}^{\infty}\frac{(4n)!(1103+26390n)}{(n!)^{4}396^{4n}}</math> | ||
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| − | + | ==라마누잔의 class invariants== | |
| − | + | * [[라마누잔의 class invariants]] | |
| − | + | :<math>g_{58}^2=\frac{\sqrt{29}+5}{2}</math> | |
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| − | < | + | ==메모== |
| − | + | * <math>e^{\sqrt{58}\pi}=24591257751.999999822\cdots</math> | |
| − | + | * <math>\frac{6}{5}{\phi}^2\approx{\pi}</math> | |
| + | * http://mathoverflow.net/questions/161836/ramanujans-pi-formulas-with-a-twist | ||
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| − | * | + | ==역사== |
| + | * 1910 - 라마누잔이 다음의 공식을 발견 | ||
| + | :<math>\frac{1}{\pi}= \frac{2\sqrt2}{9801}\sum_{n=0}^{\infty}\frac{(4n)!(1103+26390n)}{(n!)^{4}396^{4n}}</math> | ||
| + | * 1985 William Gosper가 이 급수를 이용하여 <math>\pi</math>값을 1700만 자리까지 계산 | ||
| + | * [[수학사 연표]] | ||
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| − | + | ==관련된 항목들== | |
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* [[산술기하평균함수(AGM)와 파이값의 계산|AGM과 파이값의 계산]] | * [[산술기하평균함수(AGM)와 파이값의 계산|AGM과 파이값의 계산]] | ||
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* [[타원함수]] | * [[타원함수]] | ||
* [[모듈라 군, j-invariant and the singular moduli|The modular group, j-invariant and the singular moduli]] | * [[모듈라 군, j-invariant and the singular moduli|The modular group, j-invariant and the singular moduli]] | ||
| − | * [[초기하급수(Hypergeometric series) | + | * [[초기하급수(Hypergeometric series)]] |
| + | * [[타원적분]] | ||
* [[숫자 163]] | * [[숫자 163]] | ||
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| − | + | ==매스매티카 파일 및 계산 리소스== | |
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| − | + | * https://docs.google.com/leaf?id=0B8XXo8Tve1cxMzQzNzVlZDAtZTgwZi00YWNiLWI3M2YtNTBkNDEzYjIyN2I4&sort=name&layout=list&num=50 | |
| + | * http://documents.wolfram.com/mathematica/Demos/Notebooks/CalculatingPi.html | ||
| + | * http://functions.wolfram.com/Constants/Pi/06/01/02/0001/ | ||
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| − | + | ==사전 형태의 자료== | |
| − | * http://en.wikipedia.org/wiki/ | + | * http://en.wikipedia.org/wiki/Ramanujan-Sato_series |
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| − | + | ==관련도서== | |
| − | + | * '''[BB1998]'''[http://www.amazon.com/PI-AGM-Analytic-Computational-Complexity/dp/047131515X Pi and the AGM] | |
| + | ** Jonathan M. Borwein, Peter B. Borwein, Wiley-Interscience (July 13, 1998) | ||
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| − | + | ==리뷰논문, 에세이, 강의노트== | |
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| − | [http:// | + | * J. M. Borwein, P. B. Borwein and D. H. Bailey [http://www.cecm.sfu.ca/organics/papers/borwein/paper/html/paper.html Ramanujan, Modular Equations, and Approximations to Pi or How to compute One Billion Digits of Pi] |
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| − | < | + | ==관련논문== |
| + | * Jesús Guillera, New proofs of Borwein-type algorithms for Pi, arXiv:1604.00193[math.NT], April 01 2016, http://arxiv.org/abs/1604.00193v1 | ||
| + | * Cooper, Shaun, and Wadim Zudilin. “Holonomic Alchemy and Series for <math>1/\pi</math>.” arXiv:1512.04608 [math], December 14, 2015. http://arxiv.org/abs/1512.04608. | ||
| + | * Osburn, Robert, and Wadim Zudilin. ‘On the (K.2) Supercongruence of Van Hamme’. arXiv:1504.01976 [math], 8 April 2015. http://arxiv.org/abs/1504.01976. | ||
| + | * Swisher, Holly. “On the Supercongruence Conjectures of van Hamme.” arXiv:1504.01028 [math], April 4, 2015. http://arxiv.org/abs/1504.01028. | ||
| + | * [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WK2-4PW5XTP-8&_user=4420&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000059607&_version=1&_urlVersion=0&_userid=4420&md5=07a10c67e340156fe912e39d39c0330a Ramanujan's series for 1/π arising from his cubic and quartic theories of elliptic functions] | ||
| + | ** Nayandeep Deka Baruaha, and Bruce C. Berndt, Journal of Mathematical Analysis and Applications, Volume 341, Issue 1, 2007 | ||
| + | * [http://www.math.rutgers.edu/%7Ezeilberg/mamarim/mamarimhtml/ramapi.html A WZ Proof of Ramanujan's Formula for Pi ] | ||
| + | ** Shalosh B. Ekhad and Doron Zeilberger, `Geometry, Analysis, and Mechanics', ed. by J.M. Rassias, World Scientific, Singapore, 1994, 107-108. | ||
| + | * [http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P62.pdf Class number three Ramanujan type series for 1/pi] | ||
| + | ** J. M. Borwein ,P. B. Borwein, Journal of Computational and Applied Mathematics (Vol.46 NO.1 / 1993) | ||
| + | * [http://www.jstor.org/stable/2325206 Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi] | ||
| + | ** J. M. Borwein, P. B. Borwein and D. H. Bailey, <cite style="line-height: 2em;">The American Mathematical Monthly</cite>, Vol. 96, No. 3 (Mar., 1989), pp. 201-219 | ||
| + | * '''[CHU88]'''Approximations and complex multiplication according to Ramanujan | ||
| + | ** D. V. Chudnovsky and G. V. Chudnovsky, Ramanujan Revisited, Academic Press Inc., Boston, (1988), p. 375-396 & p. 468-472 | ||
| + | * [http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P35.pdf Explicit Ramanujan-type approximations to pi of high order ] | ||
| + | ** J. M. Borwein, P. B. Borwein, 1987 | ||
| + | * '''[RAM1914]'''[http://www.imsc.res.in/%7Erao/ramanujan/CamUnivCpapers/Cpaper6/page1.htm Modular equations and approximations to Pi] | ||
| + | ** S. Ramanujan, Quart. J. Pure Appl. Math., (1914), vol. 45, p. 350-372 | ||
| − | * The Mountains of Pi | + | ==관련기사== |
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| + | * The Mountains of Pi | ||
** The New Yorker, 1992-3-2 | ** The New Yorker, 1992-3-2 | ||
| − | + | [[분류:원주율]] | |
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| − | + | ==메타데이터== | |
| + | ===위키데이터=== | ||
| + | * ID : [https://www.wikidata.org/wiki/Q17164696 Q17164696] | ||
| + | ===Spacy 패턴 목록=== | ||
| + | * [{'LOWER': 'ramanujan'}, {'OP': '*'}, {'LOWER': 'sato'}, {'LEMMA': 'series'}] | ||
2021년 2월 17일 (수) 04:04 기준 최신판
개요
- 라마누잔은 1914년에 다음과 같은 공식을 발표 [RAM1914]\[\frac{1}{\pi}= \frac{2\sqrt2}{9801}\sum_{n=0}^{\infty}\frac{(4n)!(1103+26390n)}{(n!)^{4}396^{4n}}\]
- Chudnovsky 형제 [CHU88]
\[\frac{426880 \sqrt{10005}}{\pi} = \sum_{k=0}^\infty \frac{(6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 (-640320)^{3k}}\!\]
정의와 미리 알아야 할 것들
\(q=e^{2\pi i \tau}\)
\(\theta_{2}(\tau)= \sum_{n=-\infty}^\infty q^{(n+\frac{1}{2})^2/2}\)
\(\theta_3(\tau)=\sum_{n=-\infty}^\infty q^{n^2/2}\)
\(\theta_{4}(\tau)= \sum_{n=-\infty}^\infty (-1)^n q^{n^2/2}\)
\(k=k(\tau)=\frac{\theta_2^2(\tau)}{\theta_3^2(\tau)}\)
\(K(k) = \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-k^2 \sin^2\theta}}\)
\(E(k) = \int_0^{\frac{\pi}{2}} \sqrt{1-k^2 \sin^2\theta}d\theta\)
\(k'=\sqrt{1-k^2}=\frac{\theta_4^2(\tau)}{\theta_3^2(\tau)}\)
\(K'(k) = K(k')\)
\(E'(k) = E(k')\)
- 위의 함수들을 이용하여, 양수 \(r\)에 대하여 다음을 정의
\(\lambda^{*}(r):=k(i\sqrt{r})\)
\(\alpha(r):=\frac{E'}{K}-\frac{\pi}{4K^2}\)
- 라마누잔의 class invariants\[g_n:=2^{-1/4}\frac{\eta(\frac{\sqrt{-n}}{2})}{\eta(\sqrt{-n})}\]
singular value function
- 타원적분이 만족시키는 르장드르 항등식 (AGM과 파이값의 계산)
\[E(k)K'(k)+E'(k)K(k)-K(k)K'(k)=\frac{\pi}{2}\label{leg}\]
- 타원적분의 성질
\[K'(\lambda^{*}(r))=\sqrt{r}K(\lambda^{*}(r))\label{ell}\]
- \ref{leg}와 \ref{ell}로부터 다음을 얻는다
\[\alpha(r)=\frac{\pi}{4K^2}-\sqrt{r}(\frac{E}{K}-1)\]
- 여기에 타원적분이 만족시키는 미분방정식
\[\frac{dK}{dk}=\frac{E-k'^2K}{kk'^2}\] 을 사용하면 \[\alpha(r)=\frac{1}{\pi}(\frac{\pi}{2K})^2-\sqrt{r}(kk'^2\frac{\dot{K}}{K}-k^2)\] 를 얻게 되고, 이를 다시 쓰면 \[\frac{1}{\pi}=\sqrt{N}k_Nk'^2_N\frac{4K\dot{K}}{\pi^2}+[\alpha(N)-\sqrt{N}k^2_N]\frac{4K^2}{\pi^2}\]
- \([\frac{2}{\pi}K(k)]^2 =m(k)F(y(k))\) 꼴로 쓰여질때, 양변을 미분하면 다음을 얻는다\[\frac{4K\dot{K}}{\pi^2}=\frac{1}{2}\dot{m}F+\frac{1}{2}m\dot{y}\dot{F}(y)\]
- 초기하급수를 다음과 같이 쓰면
\[F(y)=\sum_{n=0}^{\infty}a_ny^n\] 다음을 얻는다 \[\frac{1}{\pi}=\sum_{n=0}^\infty a_n[\frac{\sqrt{N}}{2}k{k'}^2\dot{m}+[\alpha(N)-\sqrt{N}k^2_N]m+\frac{n\sqrt{N}}{2}m\frac{\dot{y}}{y}kk'^2]y^n\]
라마누잔 파이 공식의 유도
- 아래의 prop, thm 번호는 [BB1998] 참조
- 초기하급수(Hypergeometric series) 항목의 Clausen 항등식이 중요하게 사용됨
- prop 5.6\[\frac{2}{\pi}K_s(h) = \,_2F_1(\frac{1}{4}-\frac{s}{2},\frac{1}{4}+\frac{s}{2};1;(2hh')^2)\]\[[\frac{2}{\pi}K_s(h)]^2 = \,_2F_1(\frac{1}{2}-s,\frac{1}{2}+s,\frac{1}{2};1,1;(2hh')^2)\]
- prop 5.7\[K_{1/4}(h)=(1+k^2)^{1/2}K(k)\] if \(2hh'=[\frac{g^{12}+g^{-12}}{2}]^{-1}\)
- Thm 5.6\[\frac{2}{\pi}K(k) =(1+k^2)^{-1/2} \,_2F_1(\frac{1}{8},\frac{3}{8};1;[\frac{g^{12}+g^{-12}}{2}]^{-2})\]
- Thm 5.7\[[\frac{2}{\pi}K(k)]^2 =(1+k^2)^{-1} \,_3F_2(\frac{1}{4},\frac{3}{4},\frac{1}{2};1,1;[\frac{g^{12}+g^{-12}}{2}]^{-2})\]
- (5.5.16)\[\frac{1}{\pi}=\sum_{n=0}^{\infty}\frac{(\frac{1}{4})_n(\frac{1}{2})_n(\frac{3}{4})_n}{(n!)^3}d_n(N)x_N^{2n+1}\]\[x_N=(\frac{g_N^{12}+g_N^{-12}}{2})^{-1}\]\[d_n(N)=[\frac{\alpha(N)x_N^{-1}}{1+k_N^2}-\frac{\sqrt{N}}{4}g_N^{-12}]+n\sqrt N(\frac{g_N^{12}-g_N^{-12}}{2})\]
- \(N=58\) 일 때
\[x_{58}=\frac{1}{99^2}=\frac{1}{9801},\] \[d_n(58)=(1103+26390n)2\sqrt 2\] 에서 다음을 얻는다 \[\frac{1}{\pi}= \frac{2\sqrt2}{9801}\sum_{n=0}^{\infty}\frac{(4n)!(1103+26390n)}{(n!)^{4}396^{4n}}\]
라마누잔의 class invariants
\[g_{58}^2=\frac{\sqrt{29}+5}{2}\]
메모
- \(e^{\sqrt{58}\pi}=24591257751.999999822\cdots\)
- \(\frac{6}{5}{\phi}^2\approx{\pi}\)
- http://mathoverflow.net/questions/161836/ramanujans-pi-formulas-with-a-twist
역사
- 1910 - 라마누잔이 다음의 공식을 발견
\[\frac{1}{\pi}= \frac{2\sqrt2}{9801}\sum_{n=0}^{\infty}\frac{(4n)!(1103+26390n)}{(n!)^{4}396^{4n}}\]
- 1985 William Gosper가 이 급수를 이용하여 \(\pi\)값을 1700만 자리까지 계산
- 수학사 연표
관련된 항목들
- AGM과 파이값의 계산
- 타원함수
- The modular group, j-invariant and the singular moduli
- 초기하급수(Hypergeometric series)
- 타원적분
- 숫자 163
매스매티카 파일 및 계산 리소스
- https://docs.google.com/leaf?id=0B8XXo8Tve1cxMzQzNzVlZDAtZTgwZi00YWNiLWI3M2YtNTBkNDEzYjIyN2I4&sort=name&layout=list&num=50
- http://documents.wolfram.com/mathematica/Demos/Notebooks/CalculatingPi.html
- http://functions.wolfram.com/Constants/Pi/06/01/02/0001/
사전 형태의 자료
관련도서
- [BB1998]Pi and the AGM
- Jonathan M. Borwein, Peter B. Borwein, Wiley-Interscience (July 13, 1998)
리뷰논문, 에세이, 강의노트
- J. M. Borwein, P. B. Borwein and D. H. Bailey Ramanujan, Modular Equations, and Approximations to Pi or How to compute One Billion Digits of Pi
관련논문
- Jesús Guillera, New proofs of Borwein-type algorithms for Pi, arXiv:1604.00193[math.NT], April 01 2016, http://arxiv.org/abs/1604.00193v1
- Cooper, Shaun, and Wadim Zudilin. “Holonomic Alchemy and Series for \(1/\pi\).” arXiv:1512.04608 [math], December 14, 2015. http://arxiv.org/abs/1512.04608.
- Osburn, Robert, and Wadim Zudilin. ‘On the (K.2) Supercongruence of Van Hamme’. arXiv:1504.01976 [math], 8 April 2015. http://arxiv.org/abs/1504.01976.
- Swisher, Holly. “On the Supercongruence Conjectures of van Hamme.” arXiv:1504.01028 [math], April 4, 2015. http://arxiv.org/abs/1504.01028.
- Ramanujan's series for 1/π arising from his cubic and quartic theories of elliptic functions
- Nayandeep Deka Baruaha, and Bruce C. Berndt, Journal of Mathematical Analysis and Applications, Volume 341, Issue 1, 2007
- A WZ Proof of Ramanujan's Formula for Pi
- Shalosh B. Ekhad and Doron Zeilberger, `Geometry, Analysis, and Mechanics', ed. by J.M. Rassias, World Scientific, Singapore, 1994, 107-108.
- Class number three Ramanujan type series for 1/pi
- J. M. Borwein ,P. B. Borwein, Journal of Computational and Applied Mathematics (Vol.46 NO.1 / 1993)
- Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi
- J. M. Borwein, P. B. Borwein and D. H. Bailey, The American Mathematical Monthly, Vol. 96, No. 3 (Mar., 1989), pp. 201-219
- [CHU88]Approximations and complex multiplication according to Ramanujan
- D. V. Chudnovsky and G. V. Chudnovsky, Ramanujan Revisited, Academic Press Inc., Boston, (1988), p. 375-396 & p. 468-472
- Explicit Ramanujan-type approximations to pi of high order
- J. M. Borwein, P. B. Borwein, 1987
- [RAM1914]Modular equations and approximations to Pi
- S. Ramanujan, Quart. J. Pure Appl. Math., (1914), vol. 45, p. 350-372
관련기사
- The Mountains of Pi
- The New Yorker, 1992-3-2
메타데이터
위키데이터
- ID : Q17164696
Spacy 패턴 목록
- [{'LOWER': 'ramanujan'}, {'OP': '*'}, {'LOWER': 'sato'}, {'LEMMA': 'series'}]