A new numerical method for inverse Laplace transforms used to obtain gluon distributions from the proton structure function

We recently derived a very accurate and fast new algorithm for numerically inverting the Laplace transforms needed to obtain gluon distributions from the proton structure function F(2)(gamma p) (x, Q(2)). We numerically inverted the function g(s), s being the variable in Laplace space, to G(v), where v is the variable in ordinary space. We have since discovered that the algorithm does not work if g(s) -> 0 less rapidly than 1/s as s -> infinity, e.g., as 1/s(beta) for 0 < beta < 1. In this note, we derive a new numerical algorithm for such cases, which holds for all positive and non-integer negative values of beta. The new algorithm is exact if the original function G(v) is given by the product of a power v(beta-1) and a polynomial in v. We test the algorithm numerically for very small positive beta, beta = 10(-6) obtaining numerical results that imitate the Dirac delta function delta(v). We also devolve the published MSTW2008LO gluon distribution at virtuality Q(2) = 5 GeV(2) down to the lower virtuality Q(2) = 1.69 GeV(2). For devolution, beta is negative, giving rise to inverse Laplace transforms that are distributions and not proper functions. This requires us to introduce the concept of Hadamard Finite Part integrals, which we discuss in detail.