Monthly Archives: July 2017

Calculating an auto-convolution integral by Fourier transforms

Recently I came across a forum contribution [1] where the author describes how to use the unitarity of the Fourier transform on $L^2(\RR)$ to compute the definite integral
I(t_0) \DEF \int_{-\infty}^{\infty}{\frac{\sin(\tau – t_0)}{(\tau – t_0)}\frac{\sin(\tau + t_0)}{(\tau + t_0)}\,d\tau},
where $t_0 \in \RR$ is a constant. In the comments it is also argued that one may alternatively use contour integration. I liked the article, but wondered whether the symmetry of the problem would perhaps admit a simpler approach. Here I give a third method that uses Fourier transforms and convolutions.

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