1A REAL-VALUED INTEGRAL WITH KERNEL ( , ) ()nK n x L x TO INVERT THE LAPLACE TRANSFORM

ABSTRACT: = In this paper we introduce an integral over the real line involving the Laguerre functions ()nL x for arbitrary real n. We use the orthogonality properties of Laguerre functions to get f(t) knowing its Laplace transform F(s), without using the Cauchy formula to evaluate the residues, we discuss the advantages and disadvantages of this new method and how it could be applied not only to the Laplace inverse method but also to evaluate Arithmetical functions in Number theory expressing the Laplace inverse by an integral over (0,) Keywords: = Laplace transform, Laguerre functions, integral Kernel 1