"라마누잔과 파이"의 두 판 사이의 차이

수학노트
둘러보기로 가기 검색하러 가기
 
(사용자 2명의 중간 판 71개는 보이지 않습니다)
1번째 줄: 1번째 줄:
−
<h5>간단한 소개</h5>
+
==개요==
  
−
; <math>\frac{1}{\pi}= \frac{2\sqrt2}{9801}\sum_{n=0}^{\infty}\frac{(4n)!(1103+26390n)}{(n!)^{4}396^{4n}}</math>
+
* 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>
  
−
 
+
* [[숫자 163]]
  
−
:
+
−
<math>\frac{426880 \sqrt{10005}}{\pi} = \sum_{k=0}^\infty \frac{(6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 (-640320)^{3k}}\!</math>
 
  
−
 
+
  
 +
==정의와 미리 알아야 할 것들==
  
−
 
+
* [[자코비 세타함수]], [[라마누잔의 class invariants]], [[타원적분]] 참조
  
−
<h5>하위주제들</h5>
+
<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>
  
−
==== 하위페이지 ====
+
<math>k=k(\tau)=\frac{\theta_2^2(\tau)}{\theta_3^2(\tau)}</math>
  
−
* [[1964250|0 토픽용템플릿]]<br>
+
<math>K(k) = \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-k^2 \sin^2\theta}}</math>
−
** [[2060652|0 상위주제템플릿]]<br>
 
  
−
 
+
<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>
  
−
<h5>재미있는 사실</h5>
+
<math>K'(k) = K(k')</math>
  
−
 
+
<math>E'(k) = E(k')</math>
  
−
 
+
* 위의 함수들을 이용하여, 양수 <math>r</math>에 대하여 다음을 정의
  
−
<h5>관련된 단원</h5>
+
<math>\lambda^{*}(r):=k(i\sqrt{r})</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>
  
−
<h5>많이 나오는 질문</h5>
+
  
−
* 네이버 지식인<br>
+
   
−
** http://kin.search.naver.com/search.naver?where=kin_qna&query=
 
  
−
 
+
==singular value function ==
  
−
<h5>관련된 고교수학 또는 대학수학</h5>
+
*  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>
  
−
 
+
  
−
 
+
* <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>
  
−
<h5>관련된 다른 주제들</h5>
+
  
−
* [[산술기하평균함수(AGM)와 파이값의 계산|AGM과 파이값의 계산]]
+
−
* [[타원적분(통합됨)|타원적분]]
+
 
−
* [[모듈라 군, j-invariant and the singular moduli|The modular group, j-invariant and the singular moduli]]
+
==라마누잔 파이 공식의 유도==
−
* [[초기하급수(Hypergeometric series)|Hypergeometric functions]]
+
 
 +
* 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>
 +
 
 +
 +
 
 +
* <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> 
 +
 
 +
 +
 
 +
 +
 
 +
==라마누잔의 class invariants==
 +
 
 +
* [[라마누잔의 class invariants]]
 +
:<math>g_{58}^2=\frac{\sqrt{29}+5}{2}</math>
 +
 
 +
 +
 
 +
 +
 
 +
==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
 +
 +
 
 +
  
−
 
+
==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]]
  
−
<h5>관련도서 및 추천도서</h5>
+
 +
  
−
* [http://www.amazon.com/PI-AGM-Analytic-Computational-Complexity/dp/047131515X Pi and the AGM]<br>
+
==관련된 항목들==
−
** Jonathan M. Borwein, Peter B. Borwein
 
−
*  도서내검색<br>
 
−
** http://books.google.com/books?q=
 
−
** http://book.daum.net/search/contentSearch.do?query=
 
−
*  도서검색<br>
 
−
** http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
 
−
** http://book.daum.net/search/mainSearch.do?query=
 
  
−
 
+
* [[산술기하평균함수(AGM)와 파이값의 계산|AGM과 파이값의 계산]]
 +
* [[타원함수]]
 +
* [[모듈라 군, j-invariant and the singular moduli|The modular group, j-invariant and the singular moduli]]
 +
* [[초기하급수(Hypergeometric series)]]
 +
* [[타원적분]]
 +
* [[숫자 163]]
  
−
<h5>참고할만한 자료</h5>
+
  
−
* 공식을 구현한 매쓰매티카 파일<br>
+
   
−
** [[3006616/attachments/1360956|ramanujan_pi.nb]]
 
−
* [http://www.jstor.org/stable/2325206 Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi]<br>
 
−
** J. M. Borwein, P. B. Borwein and D. H. Bailey
 
−
** <cite>The American Mathematical Monthly</cite>, Vol. 96, No. 3 (Mar., 1989), pp. 201-219
 
−
* [http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P62.pdf Class number three Ramanujan type series for 1/pi]<br>
 
−
** J. M. Borwein ,P. B. Borwein
 
−
** Journal of Computational and Applied Mathematics (Vol.46 NO.1 / 1993)
 
−
* [http://www.imsc.res.in/%7Erao/ramanujan/CamUnivCpapers/Cpaper6/page1.htm Modular equations and approximations to Pi]<br>
 
−
** S. Ramanujan
 
−
** Quart. J. Pure Appl. Math., (1914), vol. 45, p. 350-372
 
−
*  Approximations and complex multiplication according to Ramanujan<br>
 
−
** D. V. Chudnovsky and G. V. Chudnovsky,
 
−
** Ramanujan Revisited, Academic Press Inc., Boston, (1988), p. 375-396 & p. 468-472.
 
−
* [http://www.math.rutgers.edu/%7Ezeilberg/mamarim/mamarimhtml/ramapi.html A WZ Proof of Ramanujan's Formula for Pi ]<br>
 
−
** 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/P35.pdf Explicit Ramanujan-type approximations to pi of high order ]<br>
 
−
** J. M. Borwein, P. B. Borwein
 
−
** 1987
 
−
* [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]<br>
 
−
** Nayandeep Deka Baruaha, and Bruce C. Berndt
 
−
** Journal of Mathematical Analysis and Applications, Volume 341, Issue 1, 2007
 
  
−
[http://ko.wikipedia.org/wiki/http://en.wikipedia.org/wiki/ ]
+
==매스매티카 파일 및 계산 리소스==
  
−
* http://ko.wikipedia.org/wiki/
+
* https://docs.google.com/leaf?id=0B8XXo8Tve1cxMzQzNzVlZDAtZTgwZi00YWNiLWI3M2YtNTBkNDEzYjIyN2I4&sort=name&layout=list&num=50
−
* http://en.wikipedia.org/wiki/
+
* http://documents.wolfram.com/mathematica/Demos/Notebooks/CalculatingPi.html
 +
* http://functions.wolfram.com/Constants/Pi/06/01/02/0001/
  
−
 
 
  
−
<h5>관련기사</h5>
+
==사전 형태의 자료==
 +
* http://en.wikipedia.org/wiki/Ramanujan-Sato_series
  
−
*  The Mountains of Pi<br>
+
==관련도서==
−
** The New Yorker
 
−
*  네이버 뉴스 검색 (키워드 수정)<br>
 
−
** [http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=%EB%9D%BC%EB%A7%88%EB%88%84%EC%9E%94 http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=라마누잔]
 
−
** [http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=%ED%8C%8C%EC%9D%B4 http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=파이]
 
−
** http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
 
−
** http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
 
−
** http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=
 
  
−
 
+
* '''[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)
  
−
 
+
 +
  
−
<h5>블로그</h5>
+
==리뷰논문, 에세이, 강의노트==
  
−
* 구글 블로그 검색 http://blogsearch.google.com/blogsearch?q=
+
* 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]
−
* 트렌비 블로그 검색 http://www.trenb.com/search.qst?q=
 
  
−
 
+
  
−
<h5>이미지 검색</h5>
+
  
−
* http://commons.wikimedia.org/w/index.php?title=Special%3ASearch&search=
+
==관련논문==
−
* http://images.google.com/images?q=
+
* Jesús Guillera, New proofs of Borwein-type algorithms for Pi, arXiv:1604.00193[math.NT], April 01 2016, http://arxiv.org/abs/1604.00193v1
−
* [http://www.artchive.com/ http://www.artchive.com]
+
* 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
  
−
 
+
==관련기사==
  
−
<h5>동영상</h5>
+
*  The Mountains of Pi
 +
** The New Yorker, 1992-3-2
 +
[[분류:원주율]]
  
−
* http://www.youtube.com/results?search_type=&search_query=
+
==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

\(\lambda^{*}(r):=k(i\sqrt{r})\)

\(\alpha(r):=\frac{E'}{K}-\frac{\pi}{4K^2}\)



singular value function

\[E(k)K'(k)+E'(k)K(k)-K(k)K'(k)=\frac{\pi}{2}\label{leg}\]

  • 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



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



매스매티카 파일 및 계산 리소스


사전 형태의 자료

관련도서

  • [BB1998]Pi and the AGM
    • Jonathan M. Borwein, Peter B. Borwein, Wiley-Interscience (July 13, 1998)



리뷰논문, 에세이, 강의노트



관련논문

관련기사

  • The Mountains of Pi
    • The New Yorker, 1992-3-2

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'ramanujan'}, {'OP': '*'}, {'LOWER': 'sato'}, {'LEMMA': 'series'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=라마누잔과_파이&oldid=52248"