라마누잔의 정적분

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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.



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