자코비 세타함수

간단한 소개

• 세타함수의 정의 (spectral decomposition of heat kernel)
  \(\theta(\tau)= \sum_{n=-\infty}^\infty \exp(\pi i n^2\tau)\)
  

 

\(\theta_2^4+\theta_4^4=\theta_3^4\)

 

 

세타함수의 Modularity

\(\theta(-\frac{1}{\tau})=\sqrt{\frac{\tau}{i}} \theta({\tau})\)

\(\tau=iy, y>0\) 으로 쓰면,

\(\theta(\frac{i}{y})=\sqrt{y} \theta({iy)\)

 

Triple product 공식

\(\sum_{n=-\infty}^\infty z^{n}q^{n^2}= \prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 + zq^{2m-1}\right) \left( 1 + z^{-1}q^{2m-1}\right)\)

\(z=1\) 인 경우

\(\sum_{n=-\infty}^\infty q^{n^2}= \prod_{m=1}^\infty \left( 1 - q^{2m}\right) \left( 1 + q^{2m-1}\right)^2\)

 

 

세타함수와 타원적분

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

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

\(\theta_3(q)+\theta_4(q)=2\theta_3(q^4)\)

 

 

\(\frac{\theta_3^2(q)+\theta_4^2(q)}{2}=\theta_3(q^2)\)

\(\frac{\theta_3^2(q)+\theta_4^2(q)}{2}=\theta_3(q^2)\)

 

 

 

 

 

 

 

관련된 다른 주제들

• 타원함수
• 이차형식

 

표준적인 도서 및 추천도서

• Brief Introduction to Theta Functions
  
  • BELLMAN, RICHARD
• Tata Lectures on Theta I,II,III
  
  • David Mumford

위키링크

• http://en.wikipedia.org/wiki/Theta_functions

 

참고할만한 자료

• Applications of Theta Functions to Arithmetic
  
  • G. D. Nichols
  • The American Mathematical Monthly, Vol. 45, No. 6 (Jun. - Jul., 1938), pp. 363-368
• A First Course in Modular Forms (Graduate Texts in Mathematics)
  
  • Fred Diamond and Jerry Shurman
  • 18-19p four_square_theorem_and_theta_funtion.pdf
• Karl Gustav Jacob Jacobi
  
  • Jacobi's Four Square Theorem. (Also available in postscript format [11 pages].) [CONSTRUCTION IN PROGRESS]