- #1
mnb96
- 715
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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:
[tex]\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)[/tex]
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:
[tex]\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[/tex]
because Gaussians do not have a primitive function that allows computing that integral between [0,1].
Any hint?
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:
[tex]\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)[/tex]
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:
[tex]\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[/tex]
because Gaussians do not have a primitive function that allows computing that integral between [0,1].
Any hint?
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