The Laplace transform of the psi function

Abstract:

An expression for the Laplace transform of the psi function \begin{equation} L(a):=\int _{0}^{\infty }e^{-at}\psi (t+1) dt\nonumber \\ \end{equation} is derived using two different methods. It is then applied to evaluate the definite integral \begin{equation} M(a)=\frac {4}{\pi } \int _{0}^{\infty }\frac {x^2 dx}{x^2+\ln ^{2}(2e^{-a}\cos x)},\nonumber \\ \end{equation} for $a>\ln 2$ and to resolve a conjecture posed by Olivier Oloa.