On the isolation of zeroes of an analytic function

Douglas S. Bridges

Vol. 96 (1981), No. 1, 13–22

DOI: 10.2140/pjm.1981.96.13

Abstract

The purpose of this paper is to describe extensions of the work of Errett Bishop on the location of zeroes of complex-valued analytic functions. The main result deals with the number of zeroes of an analytic function f near the boundary of a closed disc well contained in the domain of f. A particular consequence of this result is the following theorem.

Let f be analytic and not identically zero on a connected open subset U of C, K a compact set well contained in U, and 𝜀 > 0. Then either inf{|f(z)| : z ∈ K} > 0 or there exist finitely many points z₁,⋯,z_n of U and an analytic function g on U such that

inf{|g(z)| : z ∈ K} > 0 and d(z_k,K) < 𝜀 for each k.

The paper is written entirely within the framework of Bishop’s constructive mathematics.

Mathematical Subject Classification 2000

Primary: 03F65

Secondary: 30C15

Milestones

Received: 30 November 1979

Revised: 14 March 1980

Published: 1 September 1981