"라마누잔의 class invariants"의 두 판 사이의 차이

2009년 10월 27일 (화) 10:15 판

라마누잔이 많은 계산 결과를 남겨놓은 분야
class field theory에서 중요한 역할을 함
\(G_n:=2^{-1/4}f(\sqrt{-n})\)
\(g_n:=2^{-1/4}f_1(\sqrt{-n})\)

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

자코비 세타함수
[[자코비 세타함수|]]\(\theta_{2}(\tau)= \sum_{n=-\infty}^\infty q^{(n+\frac{1}{2})^2/2}\)
\(\theta_3(\tau)=\sum_{n=-\infty}^\infty q^{n^2/2}\)
\(\theta_{4}(\tau)= \sum_{n=-\infty}^\infty (-1)^n q^{n^2/2}\)
모듈라 군, j-invariant and the singular moduli

[[모듈라 군, j-invariant and the singular moduli|]]
\(k=k(\tau)=\frac{\theta_2^2(\tau)}{\theta_3^2(\tau)}\)
\(k'=\sqrt{1-k^2}=\frac{\theta_4^2(\tau)}{\theta_3^2(\tau)}\)

베버(Weber) 모듈라 함수
[[베버(Weber) 모듈라 함수|]]
\(f(\tau)=\frac{e^{-\frac{\pi i}{24}}\eta(\frac{\tau+1}{2})}{\eta(\tau)}=q^{-1/48} \prod_{n=1}^{\infty} (1+q^{n-\frac{1}{2}})\)
\(f_1(\tau)=\frac{\eta(\frac{\tau}{2})}{\eta(\tau)}=q^{-1/48} \prod_{n=1}^{\infty} (1-q^{n-\frac{1}{2}})\)
\(f_2(\tau)=\sqrt{2}\frac{\eta(2\tau)}{\eta(\tau)}=\sqrt{2}q^{1/24} \prod_{n=1}^{\infty} (1+q^{n})\)

\(G_{25}=\frac{\sqrt{5}+1}{2}\)

\(g_{10}=\sqrt{\frac{\sqrt{5}+1}{2}}\)

\(g_{58}^2=\frac{\sqrt{29}+5}{2}\)

\(G_n:=(2kk')^{-1/12}=2^{-1/4}f(\sqrt{-n})\)

\(g_n:=(\frac{k'(\sqrt{-n})^2}{2k(\sqrt{-n})})^{1/12}=2^{-1/4}f_1(\sqrt{-n})\)

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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%;">이 항목의 스프링노트 원문주소</h5>
+* [[#]]<br>
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-<h5>정의</h5>
+<h5>필요한 정의</h5>
 <math>q=e^{2\pi i \tau}</math>
-* [[자코비 세타함수]][[자코비 세타함수|]]<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><math>\theta_{4}(\tau)= \sum_{n=-\infty}^\infty (-1)^n q^{n^2/2}</math><br>
+* [[자코비 세타함수]]<br>[[자코비 세타함수|]]<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><math>\theta_{4}(\tau)= \sum_{n=-\infty}^\infty (-1)^n q^{n^2/2}</math><br>
 * [[모듈라 군, j-invariant and the singular moduli]]<br>  <br>[[모듈라 군, j-invariant and the singular moduli|]]<br><math>k=k(\tau)=\frac{\theta_2^2(\tau)}{\theta_3^2(\tau)}</math><br><math>k'=\sqrt{1-k^2}=\frac{\theta_4^2(\tau)}{\theta_3^2(\tau)}</math><br>