  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 $g(s)\rightarrow 0$ less rapidly than $1/s$ as
$s\rightarrow\infty$, 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 {\em 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.
