The Laplace transform of the digamma function: An integral due to Glasser, Manna and Oloa

Amdeberhan, Tewodros; Espinosa, Olivier; Moll, Victor

doi:10.1090/S0002-9939-08-09300-3

The Laplace transform of the digamma function: An integral due to Glasser, Manna and Oloa
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by Tewodros Amdeberhan, Olivier Espinosa and Victor H. Moll PDF

Proc. Amer. Math. Soc. 136 (2008), 3211-3221 Request permission

Abstract:

The definite integral \begin{equation*} M(a):= \frac {4}{\pi } \int _{0}^{\pi /2} \frac {x^{2} dx } {x^{2} + \ln ^{2}( 2 e^{-a} \cos x ) }\end{equation*} is related to the Laplace transform of the digamma function \begin{equation*} L(a) := \int _{0}^{\infty } e^{-a s} \psi (s+1) ds, \end{equation*} by $M(a) = L(a) + \gamma /a$ when $a > \ln 2$. Certain analytic expressions for $M(a)$ in the complementary range, $0 < a \leq \ln 2$, are also provided.

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