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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 $F_2^{γp}(x,Q^2)$. 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 $g(s)\rightarrow 0$ less rapidly than $1/s$ as $s\rightarrow\infty$, e.g., as $1/s^β$ for $0<β<1$. In this note, we derive a new numerical algorithm for such cases, which holds for all positive and non-integer negative values of $β$. The new algorithm is {\em exact} if the original function $G(v)$ is given by the product of a power $v^{β-1}$ and a polynomial in $v$. We test the algorithm numerically for very small positive $β$, $β=10^{-6}$ obtaining numerical results that imitate the Dirac delta function $δ(v)$. 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, $ β$ 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.