An integral that counts the zeros of a function

TL;DR

This paper introduces an integral formula involving a function and its derivatives to count the zeros of the function on an interval, enabling practical computation through numerical approximation.

Contribution

It presents a novel integral-based method for counting zeros of functions, including their multiplicities, using only evaluations of the function and its derivatives.

Findings

Integral formula accurately counts zeros of functions.

Numerical approximation via trapezoidal rule is effective.

Method distinguishes zeros by multiplicity.

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 $f, f'$ and $f''$. 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 $f,f'$ and $f''$. A variant of the integral even allows to determine the number of the zeros broken down by their multiplicity.