라마누잔과 파이

간단한 소개

\(\frac{1}{\pi}= \frac{2\sqrt2}{9801}\sum_{n=0}^{\infty}\frac{(4n)!(1103+26390n)}{(n!)^{4}396^{4n}}\)

 

\[\frac{426880 \sqrt{10005}}{\pi} = \sum_{k=0}^\infty \frac{(6k)! (13591409 + 545140134k)}{(3k)!(k!)^3 (-640320)^{3k}}\!\]

 

 

 

 

 

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관련된 다른 주제들

• AGM과 파이값의 계산
• 타원적분
• The modular group, j-invariant and the singular moduli
• Hypergeometric functions

 

관련도서 및 추천도서

• Pi and the AGM
  
  • Jonathan M. Borwein, Peter B. Borwein
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참고할만한 자료

• 공식을 구현한 매쓰매티카 파일
  
  • ramanujan_pi.nb
• 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
• 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)
• Modular equations and approximations to Pi
  
  • S. Ramanujan
  • Quart. J. Pure Appl. Math., (1914), vol. 45, p. 350-372
• 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.
• 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.
• Explicit Ramanujan-type approximations to pi of high order 
  
  • J. M. Borwein, P. B. Borwein
  • 1987
• 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

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관련기사

• The Mountains of Pi
  
  • The New Yorker
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