Integrals of Distorted Gaussian Functions

Abstract

Series solutions involving Gaussian-smoothed sources may involve integrating a slightly-distorted Gaussian function, which can be expressed as a Gaussian multiplied by a polynomial or series. The 1D Gaussian function centred at $x = 0$ is $f_{1,σ}(x)$ = $\frac{1} {σ\sqrt{2π}}$ $e^{-x^{2}/2σ^{2}}$ and obeys $\int$$^{\infty}_{-\infty}$ $f_{1,σ}(x) dx =$