"라마누잔과 파이"의 두 판 사이의 차이

2009년 4월 25일 (토) 16:07 판

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

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

공식을 구현한 매쓰매티카 파일
- ramanujan_pi.nb
http://documents.wolfram.com/mathematica/Demos/Notebooks/CalculatingPi.html 참고
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

@@ 61번째 줄: / 61번째 줄: @@
 * [[모듈라 군, j-invariant and the singular moduli|The modular group, j-invariant and the singular moduli]]
 * [[초기하급수(Hypergeometric series)|Hypergeometric functions]]
+* [[숫자 163]]
@@ 81번째 줄: / 82번째 줄: @@
 *  공식을 구현한 매쓰매티카 파일<br>
 ** [[3006616/attachments/1360956|ramanujan_pi.nb]]
-* d
+* http://documents.wolfram.com/mathematica/Demos/Notebooks/CalculatingPi.html 참고
 * [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