"라마누잔과 파이"의 두 판 사이의 차이

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<h5>간단한 소개</h5>
 
<h5>간단한 소개</h5>
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* 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>
 
; <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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: <math>\frac{426880 \sqrt{10005}}{\pi} = \sum_{k=0}^\infty \frac{(6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 (-640320)^{3k}}\!</math>
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<math>\frac{426880 \sqrt{10005}}{\pi} = \sum_{k=0}^\infty \frac{(6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 (-640320)^{3k}}\!</math>
  
 
 
 
 
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* [http://www.jstor.org/stable/2325206 Ramanujan, Modular Equations, and Approximations to Pi or How to Compute One Billion Digits of Pi]<br>
 
* [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 style="line-height: 2em;">The American Mathematical Monthly</cite>, Vol. 96, No. 3 (Mar., 1989), pp. 201-219
 
** 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
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*  Approximations and complex multiplication according to Ramanujan<br>
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*  [ChUApproximations 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.
 
** 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 ]<br>
 
* [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
 
** J. M. Borwein, P. B. Borwein, 1987
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* [http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper6/page1.htm Modular equations and approximations to Pi]<br>
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* '''[RAM1914]'''[http://www.imsc.res.in/~rao/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
 
** S. Ramanujan, Quart. J. Pure Appl. Math., (1914), vol. 45, p. 350-372
  

2009년 8월 14일 (금) 11:07 판

간단한 소개
  • 라마누잔은 1914년에 다음과 같은 공식을 발표 [RAM1914]
\(\frac{1}{\pi}= \frac{2\sqrt2}{9801}\sum_{n=0}^{\infty}\frac{(4n)!(1103+26390n)}{(n!)^{4}396^{4n}}\)
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\(\frac{426880 \sqrt{10005}}{\pi} = \sum_{k=0}^\infty \frac{(6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 (-640320)^{3k}}\!\)

 

정의

 

\(\theta_{4}(\tau)= \sum_{n=-\infty}^\infty (-1)^n q^{n^2/2}\)

\(k=k(\tau)=\frac{\theta_2^2(\tau)}{\theta_3^2(\tau)}\)

\(K(k) = \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-k^2 \sin^2\theta}}\)

\(E(k) = \int_0^{\frac{\pi}{2}} \sqrt{1-k^2 \sin^2\theta}}d\theta}{\)

\(k'=\sqrt{1-k^2}=\frac{\theta_4^2(\tau)}{\theta_3^2(\tau)}\)

\(K'(k) = K(k')\)

\(E'(k) = E(k')\)

  • 위의 함수들을 이용하여, 양수 \(r\)에 대하여 다음을 정의

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

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

 

 
  • [BB1998]  (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
    \(x_{58}=\frac{1}{99^2}=\frac{1}{9801}\), \(d_n(58)=(1103+26390n)2\sqrt 2\) 이므로 다음을 얻는다
    \(\frac{1}{\pi}= \frac{2\sqrt2}{9801}\sum_{n=0}^{\infty}\frac{(4n)!(1103+26390n)}{(n!)^{4}396^{4n}}\)
     

 

 

라마누잔의 class invariants

 

 

재미있는 사실
  • \(e^{\sqrt{58}\pi}=24591257751.999999822\cdots\)

 

 


역사
  • Around 1910, the Indian mathematician Srinivasa Ramanujan discovered the formula
\(\frac{1}{\pi}= \frac{2\sqrt2}{9801}\sum_{n=0}^{\infty}\frac{(4n)!(1103+26390n)}{(n!)^{4}396^{4n}}\)
  • William Gosper used this series in 1985 to compute the first 17 million digits of \(\pi\).

 

관련된 고교수학 또는 대학수학

 

 

관련된 다른 주제들

 

관련도서 및 추천도서

 

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