로저스-라마누잔 항등식

간단한 소개

라마누잔이 하디에게 보낸 편지에는 다음과 같은 공식이 포함되어 있음

\(\cfrac{1}{1 + \cfrac{e^{-2\pi}}{1 + \cfrac{e^{-4\pi}}{1+\dots}}} = \left({\sqrt{5+\sqrt{5}\over 2}-{\sqrt{5}+1\over 2}}\right)e^{2\pi/5} = e^{2\pi/5}\left({\sqrt{\varphi\sqrt{5}}-\varphi}\right) = 0.9981360\dots\)

 

\(\varphi\) 는 황금비

 

 

 

로저스-라마누잔 연분수

\(1+\cfrac{q}{1+\cfrac{q^2}{1+\cfrac{q^3}{1+\cdots}}} = \frac{G(q)}{H(q)}=1+q -q^3 +q^5-\cdots\)

 

 

 

 

 

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관련된 다른 주제들

• The modular group, j-invariant and the singular moduli
• 오차방정식과 정이십면체

 

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참고할만한 자료

• Continued fractions and modular functions
  
  • W. Duke
  • Bull. Amer. Math. Soc. 42 (2005), 137-162.
• Explicit evaluations of the Rogers-Ramanujan continued fraction.
  
  • Berndt, B.C,Chan, H.H.,Zhang, L.-C.
  • Journal für die reine und angewandte Mathematik 480, 1996
• Ramanujan’s formulas for the explicit evaluation of the Rogers–Ramanujan continued fraction and theta-functions
  
  • Soon-Yi Kang
  • ACTA ARITHMETICA XC.1 (1999)
• Watson, G. N.
  
  • Theorems Stated by Ramanujan (IX): Two Continued Fractions.
  • Theorems Stated by Ramanujan (VII): Theorems on a Continued Fraction.
• http://ko.wikipedia.org/wiki/연분수
• http://en.wikipedia.org/wiki/Rogers%E2%80%93Ramanujan_continued_fraction
• http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=ramanujan's
• 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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블로그

• 수학과 대학원생이 되면 좋은점 - 라마누잔 이야기
  
  • 피타고라스의 창, 2009-6-24
    
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