라마누잔의 class invariants

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

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

 

 

 

정의

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

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

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

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

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

 

 

 

 

 

 

 

 

 

 

관련된 다른 주제들

• 라마누잔과 파이

 

관련도서 및 추천도서

• Ramanujan's Notebooks: V
  
  • Bruce C. Berndt
• 도서내검색
  
  • http://books.google.com/books?q=
  • http://book.daum.net/search/contentSearch.do?query=
• 도서검색
  
  • http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
  • http://book.daum.net/search/mainSearch.do?query=

 

참고할만한 자료

• Ramanujan's Most Singular Modulus
  
  • Mark B. Villarino, Arxiv, 2003-8
• Ramanujan’s class invariants and cubic continued fraction
  
  • BC Berndt, HH Chan, LC Zhang, ACTA ARITHMETICA LXXIII.1 (1995)
• Ramanujan's class invariants, Kronecker's limit formula, and modular equations
  
  • BC Berndt, HH Chan, LC Zhang, Transactions of the American Mathematical Society, 1997
• RAMANUJAN–WEBER CLASS INVARIANT Gn AND WATSON'S EMPIRICAL PROCESS
  
  • HH Chan, Journal of the London Mathematical Society, 1998
• Ramanujan and the modular j-invariant
  
  • BC Berndt, HH Chan, Canadian Mathematical Bulletin, 1999
• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/
• http://scholar.google.com/scholar?q=ramanujan%27s+class+invariants&hl=ko&lr=&start=10&sa=N
• http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
• http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
• 다음백과사전 http://enc.daum.net/dic100/search.do?q=

 

 

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