Vol. 96, No. 1, 1981

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On the isolation of zeroes of an analytic function

Douglas S. Bridges

Vol. 96 (1981), No. 1, 13–22

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,,zn of U and an analytic function g on U such that

f (z) = (z − z1)⋅⋅⋅(z − zn)g(z) (z ∈ U ),

inf{|g(z)| : z K} > 0 and d(zk,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
Received: 30 November 1979
Revised: 14 March 1980
Published: 1 September 1981
Douglas S. Bridges