An integral that counts the zeros of a function

Norbert Hungerbühler 1  and Micha Wasem 2
  • 1 Department of Mathematics, ETH Zürich, Rämistrasse 101, 8092, Zürich, Switzerland
  • 2 HTA Freiburg, HES-SO University of Applied Sciences and Arts Western Switzerland, Pérolles 80, 1700, Freiburg, Switzerland


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.

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