"라마누잔과 파이"의 두 판 사이의 차이

수학노트
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(사용자 2명의 중간 판 44개는 보이지 않습니다)
1번째 줄: 1번째 줄:
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<h5>간단한 소개</h5>
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==개요==
  
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* 라마누잔은 1914년에 다음과 같은 공식을 발표 '''[RAM1914]'''
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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>
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*
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* Chudnovsky 형제 '''[CHU88]'''
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*; <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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*  
 
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*; Chudnovsky 형제  '''[CHU88]'''
 
  
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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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* [[숫자 163]]
  
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* [[숫자 163]]
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==정의와 미리 알아야 할 것들==
  
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* [[자코비 세타함수]], [[라마누잔의 class invariants]], [[타원적분]] 참조
  
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<math>q=e^{2\pi i \tau}</math>
  
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<h5>정의와 미리 알아야 할 것들</h5>
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<math>\theta_{2}(\tau)= \sum_{n=-\infty}^\infty q^{(n+\frac{1}{2})^2/2}</math>
  
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* [[타원적분(통합됨)|타원적분]] , [[자코비 세타함수]], [[라마누잔의 class invariants]], [[초기하급수(Hypergeometric series)]] 항목 참조<br><math>q=e^{2\pi i \tau}</math><br><math>\theta_{2}(\tau)= \sum_{n=-\infty}^\infty q^{(n+\frac{1}{2})^2/2}</math><br><math>\theta_3(\tau)=\sum_{n=-\infty}^\infty q^{n^2/2}</math><br>
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<math>\theta_3(\tau)=\sum_{n=-\infty}^\infty q^{n^2/2}</math>
  
 
<math>\theta_{4}(\tau)= \sum_{n=-\infty}^\infty (-1)^n q^{n^2/2}</math>
 
<math>\theta_{4}(\tau)= \sum_{n=-\infty}^\infty (-1)^n q^{n^2/2}</math>
25번째 줄: 27번째 줄:
 
<math>K(k) = \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-k^2 \sin^2\theta}}</math>
 
<math>K(k) = \int_0^{\frac{\pi}{2}} \frac{d\theta}{\sqrt{1-k^2 \sin^2\theta}}</math>
  
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<math>E(k) = \int_0^{\frac{\pi}{2}} \sqrt{1-k^2 \sin^2\theta}}d\theta}{</math>
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<math>E(k) = \int_0^{\frac{\pi}{2}} \sqrt{1-k^2 \sin^2\theta}d\theta</math>
  
 
<math>k'=\sqrt{1-k^2}=\frac{\theta_4^2(\tau)}{\theta_3^2(\tau)}</math>
 
<math>k'=\sqrt{1-k^2}=\frac{\theta_4^2(\tau)}{\theta_3^2(\tau)}</math>
33번째 줄: 35번째 줄:
 
<math>E'(k) = E(k')</math>
 
<math>E'(k) = E(k')</math>
  
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* 위의 함수들을 이용하여, 양수 <math>r</math>에 대하여 다음을 정의
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* 위의 함수들을 이용하여, 양수 <math>r</math>에 대하여 다음을 정의
  
 
<math>\lambda^{*}(r):=k(i\sqrt{r})</math>
 
<math>\lambda^{*}(r):=k(i\sqrt{r})</math>
39번째 줄: 41번째 줄:
 
<math>\alpha(r):=\frac{E'}{K}-\frac{\pi}{4K^2}</math>
 
<math>\alpha(r):=\frac{E'}{K}-\frac{\pi}{4K^2}</math>
  
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* [[라마누잔의 class invariants]]:<math>g_n:=2^{-1/4}\frac{\eta(\frac{\sqrt{-n}}{2})}{\eta(\sqrt{-n})}</math>
  
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<h5>singular value function </h5>
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*  타원적분이 만족시키는 르장드르 항등식<br>  <math>E(k)K'(k)+E'(k)K(k)-K(k)K'(k)=\frac{\pi}{2}</math> ([[산술기하평균함수(AGM)와 파이값의 계산|AGM과 파이값의 계산]])<br>
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==singular value function ==
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*  타원적분의 성질 <br><math>K'(\lambda^{*}(r))=\sqrt{r}K(\lambda^{*}(r))</math><br>
 
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*  위의 둘을 사용하여 다음을 얻는다<br><math>\alpha(r)=\frac{\pi}{4K^2}-\sqrt{r}(\frac{E}{K}-1)</math> <br>
 
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* 여기에 타원적분이 만족시키는 미분방정식
 
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*  
 
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* <math>\alpha(r)=\frac{1}{\pi}(\frac{\pi}{2K})^2-\sqrt{r}(kk'^2\frac{\.K}{K}-k^2)</math> <br>
 
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* <math>\frac{1}{\pi}=\sqrt{N}k_Nk'_N^2\frac{4K\.K}{\pi^2}+[\alpha(N)-\sqrt{N}k^2_N]\frac{4K^2}{\pi^2}</math>
 
  
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*  타원적분이 만족시키는 르장드르 항등식 ([[산술기하평균함수(AGM)와 파이값의 계산|AGM과 파이값의 계산]])
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:<math>E(k)K'(k)+E'(k)K(k)-K(k)K'(k)=\frac{\pi}{2}\label{leg}</math>
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*  타원적분의 성질
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:<math>K'(\lambda^{*}(r))=\sqrt{r}K(\lambda^{*}(r))\label{ell}</math>
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*  \ref{leg}와 \ref{ell}로부터 다음을 얻는다
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:<math>\alpha(r)=\frac{\pi}{4K^2}-\sqrt{r}(\frac{E}{K}-1)</math>
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*  여기에 타원적분이 만족시키는 미분방정식
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:<math>\frac{dK}{dk}=\frac{E-k'^2K}{kk'^2}</math> 을 사용하면
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:<math>\alpha(r)=\frac{1}{\pi}(\frac{\pi}{2K})^2-\sqrt{r}(kk'^2\frac{\dot{K}}{K}-k^2)</math> 를 얻게 되고, 이를 다시 쓰면
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:<math>\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}</math>
  
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* <math>[\frac{2}{\pi}K(k)]^2 =m(k)F(y(k))</math> 꼴로 쓰여질때, 양변을 미분하면 다음을 얻는다<br><math>\frac{4K\.K}{\pi^2}=\frac{1}{2}\.mF+\frac{1}{2}m\.y\.F(y)</math><br>
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* 초기하급수를 다음과 같이 쓰면<br><math>F(y)=\sum_{n=0}^{\infty}a_ny^n</math><br>
 
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* <math>\frac{1}{\pi}=\sum_{n=0}^\infty a_n[\frac{\sqrt{N}}{2}k{k'}^2\.m+[\alpha(N)-\sqrt{N}k^2_N]m+\frac{n\sqrt{N}}{2}m\frac{\.y}{y}kk'^2]y^n</math>
 
  
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* <math>[\frac{2}{\pi}K(k)]^2 =m(k)F(y(k))</math> 꼴로 쓰여질때, 양변을 미분하면 다음을 얻는다:<math>\frac{4K\dot{K}}{\pi^2}=\frac{1}{2}\dot{m}F+\frac{1}{2}m\dot{y}\dot{F}(y)</math>
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*  초기하급수를 다음과 같이 쓰면
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:<math>F(y)=\sum_{n=0}^{\infty}a_ny^n</math>
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다음을 얻는다
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:<math>\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</math>
  
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<h5>라마누잔 파이 공식의 유도</h5>
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* 아래의 prop, thm 번호는 '''[BB1998] '''참조
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==라마누잔 파이 공식의 유도==
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*  prop 5.6<br><math>\frac{2}{\pi}K_s(h) = \,_2F_1(\frac{1}{4}-\frac{s}{2},\frac{1}{4}+\frac{s}{2};1;(2hh')^2)</math><br><math>[\frac{2}{\pi}K_s(h)]^2 = \,_2F_1(\frac{1}{2}-s,\frac{1}{2}+s,\frac{1}{2};1,1;(2hh')^2)</math><br>
 
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*  prop 5.7<br><math>K_{1/4}(h)=(1+k^2)^{1/2}K(k)</math> if <math>2hh'=[\frac{g^{12}+g^{-12}}{2}]^{-1}</math><br>
 
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*  Thm 5.6<br><math>\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})</math><br>
 
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*  Thm 5.7<br><math>[\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})</math><br>
 
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*  (5.5.16)<br><math>\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}</math><br><math>x_N=(\frac{g_N^{12}+g_N^{-12}}{2})^{-1}</math><br><math>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})</math><br>
 
  
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* 아래의 prop, thm 번호는 '''[BB1998] '''참조
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* [[초기하급수(Hypergeometric series)]] 항목의 Clausen 항등식이 중요하게 사용됨
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*  prop 5.6:<math>\frac{2}{\pi}K_s(h) = \,_2F_1(\frac{1}{4}-\frac{s}{2},\frac{1}{4}+\frac{s}{2};1;(2hh')^2)</math>:<math>[\frac{2}{\pi}K_s(h)]^2 = \,_2F_1(\frac{1}{2}-s,\frac{1}{2}+s,\frac{1}{2};1,1;(2hh')^2)</math>
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*  prop 5.7:<math>K_{1/4}(h)=(1+k^2)^{1/2}K(k)</math> if <math>2hh'=[\frac{g^{12}+g^{-12}}{2}]^{-1}</math>
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*  Thm 5.6:<math>\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})</math>
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*  Thm 5.7:<math>[\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})</math>
 +
*  (5.5.16):<math>\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}</math>:<math>x_N=(\frac{g_N^{12}+g_N^{-12}}{2})^{-1}</math>:<math>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})</math>
  
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* <math>N=58</math> 일 때<br><math>x_{58}=\frac{1}{99^2}=\frac{1}{9801}</math>, <math>d_n(58)=(1103+26390n)2\sqrt 2</math> 이므로 다음을 얻는다<br><math>\frac{1}{\pi}= \frac{2\sqrt2}{9801}\sum_{n=0}^{\infty}\frac{(4n)!(1103+26390n)}{(n!)^{4}396^{4n}}</math><br>  <br>
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* <math>N=58</math> 일 때
 +
:<math>x_{58}=\frac{1}{99^2}=\frac{1}{9801},</math>
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:<math>d_n(58)=(1103+26390n)2\sqrt 2</math> 에서 다음을 얻는다
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:<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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<h5>라마누잔의 class invariants</h5>
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==라마누잔의 class invariants==
  
 
* [[라마누잔의 class invariants]]
 
* [[라마누잔의 class invariants]]
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* <math>g_{58}^2=\frac{\sqrt{29}+5}{2}</math>
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:<math>g_{58}^2=\frac{\sqrt{29}+5}{2}</math>
  
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<h5>재미있는 사실</h5>
 
  
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==메모==
 
* <math>e^{\sqrt{58}\pi}=24591257751.999999822\cdots</math>
 
* <math>e^{\sqrt{58}\pi}=24591257751.999999822\cdots</math>
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* <math>\frac{6}{5}{\phi}^2\approx{\pi}</math>
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* http://mathoverflow.net/questions/161836/ramanujans-pi-formulas-with-a-twist
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<h5><br> 역사</h5>
 
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* Around 1910, the Indian mathematician Srinivasa Ramanujan discovered the formula<br>
 
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; <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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*  William Gosper used this series in 1985 to compute the first 17 million digits of <math>\pi</math>.<br>
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==역사==
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* 1910 - 라마누잔이 다음의 공식을 발견
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:<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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* 1985 William Gosper가 이 급수를 이용하여 <math>\pi</math>값을 1700만 자리까지 계산
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* [[수학사 연표]]
  
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<h5>관련된 고교수학 또는 대학수학</h5>
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==관련된 항목들==
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<h5>관련된 다른 주제들</h5>
 
  
 
* [[산술기하평균함수(AGM)와 파이값의 계산|AGM과 파이값의 계산]]
 
* [[산술기하평균함수(AGM)와 파이값의 계산|AGM과 파이값의 계산]]
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* [[타원적분(통합됨)|타원적분]]
 
 
* [[타원함수]]
 
* [[타원함수]]
 
* [[모듈라 군, j-invariant and the singular moduli|The modular group, j-invariant and the singular moduli]]
 
* [[모듈라 군, j-invariant and the singular moduli|The modular group, j-invariant and the singular moduli]]
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* [[초기하급수(Hypergeometric series)|Hypergeometric functions]]
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* [[초기하급수(Hypergeometric series)]]
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* [[타원적분]]
 
* [[숫자 163]]
 
* [[숫자 163]]
  
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<h5>관련도서 및 추천도서</h5>
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* '''[BB1998]'''[http://www.amazon.com/PI-AGM-Analytic-Computational-Complexity/dp/047131515X Pi and the AGM]<br>
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==매스매티카 파일 및 계산 리소스==
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** Jonathan M. Borwein, Peter B. Borwein, Wiley-Interscience (July 13, 1998)
 
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*  도서내검색<br>
 
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** http://books.google.com/books?q=
 
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** http://book.daum.net/search/contentSearch.do?query=
 
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*  도서검색<br>
 
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** http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
 
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** http://book.daum.net/search/mainSearch.do?query=
 
  
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* https://docs.google.com/leaf?id=0B8XXo8Tve1cxMzQzNzVlZDAtZTgwZi00YWNiLWI3M2YtNTBkNDEzYjIyN2I4&sort=name&layout=list&num=50
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* http://documents.wolfram.com/mathematica/Demos/Notebooks/CalculatingPi.html
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* http://functions.wolfram.com/Constants/Pi/06/01/02/0001/
  
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<h5 style="line-height: 3.428em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic', dotum, gulim, sans-serif; font-size: 1.166em; background-image: ; background-color: initial; background-position: 0px 100%;">참고할만한 자료</h5>
 
  
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* http://ko.wikipedia.org/wiki/
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==사전 형태의 자료==
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* http://en.wikipedia.org/wiki/Pi
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* http://en.wikipedia.org/wiki/Ramanujan-Sato_series
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* http://www.wolframalpha.com/input/?i=pi
 
  
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==관련도서==
  
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<h5>참고할만한 자료</h5>
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* '''[BB1998]'''[http://www.amazon.com/PI-AGM-Analytic-Computational-Complexity/dp/047131515X Pi and the AGM]
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** Jonathan M. Borwein, Peter B. Borwein, Wiley-Interscience (July 13, 1998)
  
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* 공식을 구현한 매쓰매티카 파일<br>
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** [[3006616/attachments/1360956|ramanujan_pi.nb]]
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* http://documents.wolfram.com/mathematica/Demos/Notebooks/CalculatingPi.html 참고
 
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* [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WK2-4PW5XTP-8&_user=4420&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000059607&_version=1&_urlVersion=0&_userid=4420&md5=07a10c67e340156fe912e39d39c0330a Ramanujan's series for 1/π arising from his cubic and quartic theories of elliptic functions]<br>
 
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** Nayandeep Deka Baruaha, and Bruce C. Berndt, Journal of Mathematical Analysis and Applications, Volume 341, Issue 1, 2007
 
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* [http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/ramapi.html A WZ Proof of Ramanujan's Formula for Pi ]<br>
 
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** Shalosh B. Ekhad and Doron Zeilberger,  `Geometry, Analysis, and Mechanics', ed. by J.M. Rassias, World Scientific, Singapore, 1994, 107-108.
 
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* [http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P62.pdf Class number three Ramanujan type series for 1/pi]<br>
 
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** J. M. Borwein ,P. B. Borwein, Journal of Computational and Applied Mathematics (Vol.46 NO.1 / 1993)
 
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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>
 
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** 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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* '''[CHU88]'''Approximations 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.
 
  
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* [http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P35.pdf Explicit Ramanujan-type approximations to pi of high order ]<br>
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==리뷰논문, 에세이, 강의노트==
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** J. M. Borwein, P. B. Borwein, 1987
 
−
* '''[RAM1914]'''[http://www.imsc.res.in/~rao/ramanujan/CamUnivCpapers/Cpaper6/page1.htm Modular equations and approximations to Pi]<br>
 
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** S. Ramanujan, Quart. J. Pure Appl. Math., (1914), vol. 45, p. 350-372
 
  
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[http://ko.wikipedia.org/wiki/http://en.wikipedia.org/wiki/ ]
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* J. M. Borwein, P. B. Borwein and D. H. Bailey [http://www.cecm.sfu.ca/organics/papers/borwein/paper/html/paper.html Ramanujan, Modular Equations, and Approximations to Pi or How to compute One Billion Digits of Pi]
  
−
* http://ko.wikipedia.org/wiki/
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* http://en.wikipedia.org/wiki/
 
  
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<h5>관련기사</h5>
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==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 <math>1/\pi</math>.” 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.
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* Swisher, Holly. “On the Supercongruence Conjectures of van Hamme.” arXiv:1504.01028 [math], April 4, 2015. http://arxiv.org/abs/1504.01028.
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* [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WK2-4PW5XTP-8&_user=4420&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000059607&_version=1&_urlVersion=0&_userid=4420&md5=07a10c67e340156fe912e39d39c0330a Ramanujan's series for 1/π arising from his cubic and quartic theories of elliptic functions]
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** Nayandeep Deka Baruaha, and Bruce C. Berndt, Journal of Mathematical Analysis and Applications, Volume 341, Issue 1, 2007
 +
* [http://www.math.rutgers.edu/%7Ezeilberg/mamarim/mamarimhtml/ramapi.html 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.
 +
* [http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P62.pdf 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)
 +
* [http://www.jstor.org/stable/2325206 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, <cite style="line-height: 2em;">The American Mathematical Monthly</cite>, 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
 +
* [http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P35.pdf Explicit Ramanujan-type approximations to pi of high order ]
 +
** J. M. Borwein, P. B. Borwein, 1987
 +
* '''[RAM1914]'''[http://www.imsc.res.in/%7Erao/ramanujan/CamUnivCpapers/Cpaper6/page1.htm Modular equations and approximations to Pi]
 +
** S. Ramanujan, Quart. J. Pure Appl. Math., (1914), vol. 45, p. 350-372
  
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*  The Mountains of Pi<br>
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==관련기사==
 +
 
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*  The Mountains of Pi
 
** The New Yorker, 1992-3-2
 
** The New Yorker, 1992-3-2
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*  네이버 뉴스 검색 (키워드 수정)<br>
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[[분류:원주율]]
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** [http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=%EB%9D%BC%EB%A7%88%EB%88%84%EC%9E%94 http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=라마누잔]
 
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** [http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=%ED%8C%8C%EC%9D%B4 http://news.search.naver.com/search.naver?where=news&x=0&y=0&sm=tab_hty&query=파이]
 
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<h5>블로그</h5>
 
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* 구글 블로그 검색 http://blogsearch.google.com/blogsearch?q=
 
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* 트렌비 블로그 검색 http://www.trenb.com/search.qst?q=
 
  
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<br>
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==메타데이터==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q17164696 Q17164696]
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===Spacy 패턴 목록===
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* [{'LOWER': 'ramanujan'}, {'OP': '*'}, {'LOWER': 'sato'}, {'LEMMA': 'series'}]

2021년 2월 17일 (수) 05:04 기준 최신판

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]

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



정의와 미리 알아야 할 것들

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

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

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

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

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

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

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

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

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

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

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

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

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



singular value function

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

  • 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

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



메모



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



매스매티카 파일 및 계산 리소스


사전 형태의 자료

관련도서

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



리뷰논문, 에세이, 강의노트



관련논문

관련기사

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

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'ramanujan'}, {'OP': '*'}, {'LOWER': 'sato'}, {'LEMMA': 'series'}]
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