  We present the solution to an inverse problem arising in the context of
finding a distribution function for a specific collisionless plasma
equilibrium. The inverse problem involves the solution of two integral
equations, each having the form of a Weierstrass transform. We prove that
inverting the Weierstrass transform using Hermite polynomials leads to
convergent infinite series. We also comment on the non-negativity of the
distribution function, with more detail on this in Allanson $\textit{et al.,
Journal of Plasma Physics}$, vol. 82 (03), 2016. Whilst applied to a specific
magnetic field, the inversion techniques used in this paper (as well as the
derived convergence criteria and discussion of non-negativity) are of a general
nature, and are applicable to other smooth pressure functions.
