On the Equivalence of Fourier Expansion and Poisson Summation Formula for the Series Approximation of the Exponential Function

In this short note we show the equivalence of Fourier expansion and Poisson summation approaches for the series approximation of the exponential function $\exp ({-{t^2}/4})$. The application of the Poisson summation formula is shown to reduce to that of the Fourier expansion method.