Addendum to: "A new numerical method for obtaining gluon distribution functions $G(x,Q^2)=xg(x,Q^2)$, from the proton structure function $F_2^{\gamma p}(x,Q^2)$."

Abstract

In a recent Letter entitled "A new numerical method for obtaining gluon distribution functions $G(x,Q^2)=xg(x,Q^2)$, from the proton structure function $F_2^{\gamma p}(x,Q^2)$" [arXiv:0907.4790], we derived an accurate and fast algorithm for numerically inverting Laplace transforms, which we used in obtaining gluon distributions from the proton structure function $F_2^{\gamma p}(x,Q^2)$. We inverted the function $g(s)$, where $s$ is the variable in Laplace space, to $G(v)$, where $v$ is the variable in ordinary space. Since publication, we have discovered that the algorithm does not work if $g(s)\rightarrow 0$ less rapidly than $1/s$, as $s\rightarrow\infty$. Although we require that $g(s)\rightarrow 0$ as $s\rightarrow\infty$, it can approach 0 as ${1\over s^\beta}$, with $0<\beta<1$, and still be a proper Laplace transform. In this note, we derive a new numerical algorithm for just such cases, and test it for $g(s)={\sqrt \pi\over \sqrt s} $, the Laplace transform of ${1\over \sqrt v}$.