The Laplace transform of the psi function

Abstract

An expression for the Laplace transform of the psi function L(a):=∫ 0 ∞ e -at ψ-(t +1) dt is derived using two different methods. It is then applied to evaluate the definite integral M(a)=4/π∫ 0 ∞ x 2 dx/x 2 +In 2 (2e -1 cosx) for a > ln2 and to resolve a conjecture posed by Olivier Oloa. © 2009 American Mathematical Society.