An integral that counts the zeros of a function

Abstract

Given a real function $f$ on an interval $[a,b]$ satisfying mild regularity conditions, we determine the number of zeros of $f$ by evaluating a certain integral. The integrand depends on $f,f^{\prime }$ and $f^{\prime \prime }$. In particular, by approximating the integral with the trapezoidal rule on a fine enough grid, we can compute the number of zeros of $f$ by evaluating finitely many values of $f,f^{\prime }$ and $f^{\prime \prime }$. A variant of the integral even allows to determine the number of the zeros broken down by their multiplicity.

20 pages, 1 figure, final version