Macdonald's Theorem for Analytic Functions

TL;DR

This paper reconstructs Macdonald's theorem, revealing a relationship between zeros of an analytic function and its derivative within a bounded region with constant modulus boundary.

Contribution

It provides a detailed proof of Macdonald's theorem, which is little known and clarifies its implications for zeros of analytic functions and their derivatives.

Findings

The number of zeros of an analytic function and its derivative differ by one within the specified region.

The theorem applies to functions analytic inside a bounded region with constant modulus boundary.

The proof enhances understanding of zeros distribution for analytic functions.

Abstract

A proof is reconstructed for a useful theorem on the zeros of derivatives of analytic functions due to H. M. Macdonald, which appears to be now little known. The Theorem states that, if a function $f(z)$ is analytic inside a bounded region bounded by a contour on which the modulus of $f(z)$ is constant, then the number of zeros (counted according to multiplicity) of $f(z)$ and of its derivative in the region differ by unity.