## 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 $$\label{eq:I} I(t_0) \DEF \int_{-\infty}^{\infty}{\frac{\sin(\tau – t_0)}{(\tau – t_0)}\frac{\sin(\tau + t_0)}{(\tau + t_0)}\,d\tau},$$ where $t_0 \in \RR$ is a constant. In the… Read moreCalculating an auto-convolution integral by Fourier transforms

## Continuity of eigenvalues

As a simple corollary to the result from my previous post I would like to show you how to obtain continuous dependence of the eigenvalues of a matrix on its entries. Specifically, let be the complex vector space of matrices with entries in , endowed with any norm of our… Read moreContinuity of eigenvalues

## Continuity of polynomial roots

It was recently brought up how to show that the roots of a real or complex polynomial depend continuously on the polynomial’s coefficients. Although I have used this proposition numerous times, implicitly and explicitly, I realized that I never saw a proof of it. Perhaps the most obvious approach would… Read moreContinuity of polynomial roots