라마누잔과 파이

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]

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

• 숫자 163

정의와 미리 알아야 할 것들

• 자코비 세타함수, 라마누잔의 class invariants, 타원적분 참조

\(q=e^{2\pi i \tau}\)

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

\(\theta_3(\tau)=\sum_{n=-\infty}^\infty q^{n^2/2}\)

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

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

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

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

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

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

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

• 위의 함수들을 이용하여, 양수 \(r\)에 대하여 다음을 정의

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

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

• 라마누잔의 class invariants\[g_n:=2^{-1/4}\frac{\eta(\frac{\sqrt{-n}}{2})}{\eta(\sqrt{-n})}\]

singular value function

• 타원적분이 만족시키는 르장드르 항등식 (AGM과 파이값의 계산)

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

• 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

• 라마누잔의 class invariants

\[g_{58}^2=\frac{\sqrt{29}+5}{2}\]

메모

• \(e^{\sqrt{58}\pi}=24591257751.999999822\cdots\)
• \(\frac{6}{5}{\phi}^2\approx{\pi}\)
• http://mathoverflow.net/questions/161836/ramanujans-pi-formulas-with-a-twist

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

• AGM과 파이값의 계산
• 타원함수
• The modular group, j-invariant and the singular moduli
• 초기하급수(Hypergeometric series)
• 타원적분
• 숫자 163

매스매티카 파일 및 계산 리소스

• https://docs.google.com/leaf?id=0B8XXo8Tve1cxMzQzNzVlZDAtZTgwZi00YWNiLWI3M2YtNTBkNDEzYjIyN2I4&sort=name&layout=list&num=50
• http://documents.wolfram.com/mathematica/Demos/Notebooks/CalculatingPi.html
• http://functions.wolfram.com/Constants/Pi/06/01/02/0001/

사전 형태의 자료

• http://en.wikipedia.org/wiki/Ramanujan-Sato_series

관련도서

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

리뷰논문, 에세이, 강의노트

• J. M. Borwein, P. B. Borwein and D. H. Bailey Ramanujan, Modular Equations, and Approximations to Pi or How to compute One Billion Digits of Pi

관련논문

• Jesús Guillera, New proofs of Borwein-type algorithms for Pi, arXiv:1604.00193[math.NT], April 01 2016, http://arxiv.org/abs/1604.00193v1
• Cooper, Shaun, and Wadim Zudilin. “Holonomic Alchemy and Series for $1/\pi$.” 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.
• 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

관련기사

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