라마누잔과 파이

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]

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

타원적분 , 자코비 세타함수, 라마누잔의 class invariants 항목 참조
\(q=e^{2\pi i \tau}\)
\(\theta_{2}(\tau)= \sum_{n=-\infty}^\infty q^{(n+\frac{1}{2})^2/2}\)
\(\theta_3(\tau)=\sum_{n=-\infty}^\infty q^{n^2/2}\)

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

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

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

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

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

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

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

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

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

[BB1998]
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 \(2hh'=[(g^{12}+g^{-12})/2]^{-1}\)
Thm 5.6
\(\frac{2}{\pi}K(h) =(1+k^2)^{-1/2} \,_2F_1(\frac{1}{8},\frac{3}{8};1;(gg)^2)\)
\(K_{1/4}(h)=(1+k^2)^{1/2}K(k)\)
(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
\(\frac{1}{\pi}= \frac{2\sqrt2}{9801}\sum_{n=0}^{\infty}\frac{(4n)!(1103+26390n)}{(n!)^{4}396^{4n}}\)

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 \(\pi\).

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

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