Fourier series of a train of gaussians

[mnb96]

Hello,
I am following a proof in a book, in which the author makes use of the fact (without proving it) that the Fourier serie of a train of Gaussians is given by the following relationship:

\mathcal{F} \{ \sum_{n=-\infty}^{+\infty}\frac{1}{\tau}e^\frac{-\pi(x-n)^2}{\tau^2} \} = \sum_{m=-\infty}^{+\infty}e^{-\pi\tau^2 m^2}\cos(2\pi mx)

where n is an integer. I understand that the Gaussian train is a periodic function with period=1. However I don't quite understand how to compute its Fourier integral:

\int_0^{1} \left( \sum_{n=-\infty}^{+\infty}\frac{1}{\tau}e^\frac{-\pi(x-n)^2}{\tau^2} \right)e^{-i2\pi\ mx}dx

because Gaussians do not have a primitive function that allows computing that integral between [0,1].

Any hint?

 

[fresh_42]

The question is whether integral and summation can be swapped, since beside that we only have a constant factor and two exponential functions which we can integrate. For that we need uniform convergence and integrability on $[0,1]$.