"라마누잔의 class invariants"의 두 판 사이의 차이

2009년 10월 24일 (토) 13:58 판

라마누잔이 많은 계산 결과를 남겨놓은 분야
class field theory에서 중요한 역할을 함
\(G_n:=(2kk')^{-1/12}=2^{-1/4}f(\sqrt{-n})\)
\(g_n:=(\frac{k'(i\sqrt{n})^2}{2k(i\sqrt{n})})^{1/12}=2^{-1/4}f_1(\sqrt{-n})\)

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

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

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

Ramanujan and the modular j-invariant
- BC Berndt, HH Chan, Canadian Mathematical Bulletin, 1999

@@ 8번째 줄: / 8번째 줄: @@
 *  라마누잔이 많은 계산 결과를 남겨놓은 분야<br>
-*  class field theory에서 중요한 역할을 함<br><math>G_n:=(2kk')^{-1/12}=2^{-1/4}f(\frac{\sqrt{-n}}{2})</math><br><math>g_n:=(\frac{k'(i\sqrt{n})^2}{2k(i\sqrt{n})})^{1/12}=2^{-1/4}</math><br>
+*  class field theory에서 중요한 역할을 함<br><math>G_n:=(2kk')^{-1/12}=2^{-1/4}f(\sqrt{-n})</math><br><math>g_n:=(\frac{k'(i\sqrt{n})^2}{2k(i\sqrt{n})})^{1/12}=2^{-1/4}f_1(\sqrt{-n})</math><br>
@@ 84번째 줄: / 82번째 줄: @@
 * [http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=20087 RAMANUJAN–WEBER CLASS INVARIANT Gn AND WATSON'S EMPIRICAL PROCESS]<br>
 ** HH Chan, Journal of the London Mathematical Society, 1998
+* [http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.39.4015 Ramanujan's Class Invariants With Applications To The Values Of q-Continued Fractions And Theta-Functions]<br>
+** Bruce C. Berndt ,  Heng Huat Chan ,  Liang-Cheng Zhang, 1997
 * [http://www.ams.org/tran/1997-349-06/S0002-9947-97-01738-8/S0002-9947-97-01738-8.pdf Ramanujan's class invariants, Kronecker's limit formula, and modular equations]<br>