라마누잔의 세타함수

이 항목의 스프링노트 원문주소

• 라마누잔의 세타함수

 

 

개요

 \(f(a,b) = \sum_{n=-\infty}^\infty a^{n(n+1)/2} \; b^{n(n-1)/2}\)

 자코비 삼중곱

\(f(a,b) = (-a; ab)_\infty \;(-b; ab)_\infty \;(ab;ab)_\infty\)

 

 \(\phi(q):=f(q,q)=\sum _{n=-\infty }^{\infty } q^{n^2}=(-q;q^2)^{2}_{\infty} \left(q^2;q^2\right){}_{\infty }\)

\(\psi(q):=f(q,q^{3})=\sum _{n=0}^{\infty } q^{n(n+1)/2}=\frac{\left(q^2;q^2\right){}_{\infty }}{\left(q;q^2\right){}_{\infty }}\)

\(f(-q):=f(-q,-q^{2})=(q;q)_{\infty }\)

\(\frac{f(-q^{2},-q^{2})}{f(-q)}=\frac{\left(q^2;q^4\right)^2_{\infty }\left(q^4;q^4\right){}_{\infty }}{(q;q)_{\infty }}=\left(-q;q^2\right){}_{\infty }\)

 

 

==메모

\(f(-q)=(q;q)_{\infty}\)

\(\phi(-q)=\frac{(q;q)_{\infty}}{(-q;q)_{\infty}}\)

\(\psi(-q)=\frac{(q^{2};q^{2})_{\infty}}{(-q;q^{2})_{\infty}}\)

\(\chi(-q)=(q;q^{2})_{\infty}\)

 

 

==메모

 

• Math Overflow http://mathoverflow.net/search?q=

 

 

==역사

 

• http://www.google.com/search?hl=en&tbs=tl:1&q=
• 수학사연표
•  

 

 

==메모

• http://cis.csuohio.edu/~somos/multiq.pdf

 

==관련된 항목들

 

 

수학용어번역

• 단어사전 http://www.google.com/dictionary?langpair=en%7Cko&q=
• 발음사전 http://www.forvo.com/search/
• 대한수학회 수학 학술 용어집
  
  • http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
• 남·북한수학용어비교
• 대한수학회 수학용어한글화 게시판

 

 

==사전 형태의 자료

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/Ramanujan_theta_function
• http://mathworld.wolfram.com/RamanujanThetaFunctions.html
  [1]
  
• http://www.wolframalpha.com/input/?i=
• NIST Digital Library of Mathematical Functions
• The On-Line Encyclopedia of Integer Sequences
  
  • http://www.research.att.com/~njas/sequences/?q=

 

 

==관련논문

• http://www.jstor.org/action/doBasicSearch?Query=
• http://www.ams.org/mathscinet
• http://dx.doi.org/

 

 

==관련도서