라마누잔의 정적분

개요

\(\int_{0}^{\infty}\frac{x e^{-\sqrt{5}x}}{\cosh{x}}\,dx=\frac{1}{8}(\psi^{(1)}(\frac{1+\sqrt{5}}{4})-\psi^{(1)}(\frac{3+\sqrt{5}}{4}))\)

Integrate[(x Exp[-x Sqrt[5]])/Cosh[x], {x, 0, \[Infinity]}] // FullSimplify

[%28x+Exp[-x+Sqrt[5]%29/Cosh[x],+%7Bx,+0,+[Infinity]%7D]+ http://www.wolframalpha.com/input/?i=Integrate[(x+Exp[-x+Sqrt[5]])/Cosh[x],+{x,+0,+[Infinity]}]+]

[1,%281%2Bsqrt%285%29%29/4-polygamma[1,%283%2Bsqrt%285%29%29/4]%29/8 http://www.wolframalpha.com/input/?i=(polygamma[1,(1%2Bsqrt(5))/4]-polygamma[1,(3%2Bsqrt(5))/4])/8]

\(\int_{0}^{\infty}\frac{x^{2}e^{-\sqrt{3}x}}{\sinh{x}}\,dx=-\frac{1}{4}\psi^{(2)}(\frac{1+\sqrt{3}}{4})\)

Integrate[(x^2 Exp[-x Sqrt[3]])/Sinh[x], {x, 0, \[Infinity]}] //FullSimplify

[%28x%5E2+Exp[-x+Sqrt[3]%29/Sinh[x],+%7Bx,+0,+Infinity%7D] http://www.wolframalpha.com/input/?i=integrate[(x^2+Exp[-x+Sqrt[3]])/Sinh[x],+{x,+0,+Infinity}]]

[2,%281%2Bsqrt%283%29%29/2/4 http://www.wolframalpha.com/input/?i=-polygamma[2,(1%2Bsqrt(3))/2]/4]

Berndt, B. C. and Rankin, R. A. Ramanujan: Letters and Commentary. Providence, RI: Amer. Math. Soc., 1995.

재미있는 사실

• Math Overflow
  • Ramanujan’s eccentric Integral formula
  • 연분수 http://mathoverflow.net/questions/49866/applications-of-finite-continued-fractions/[1]

관련된 항목들

• 트리감마 함수(trigamma function)

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/
• http://mathworld.wolfram.com/RamanujanContinuedFractions.html
• 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=