Necessary and Sufficient Condition for zeros of Derivative of Meromorphic and Entire Functions

TL;DR

This paper establishes a new necessary and sufficient condition for identifying zeros of derivatives of entire and meromorphic functions, offering a novel approach to classical functions like Xi, Gamma, and digamma.

Contribution

It introduces a completely new criterion for zeros of derivatives of entire and meromorphic functions, enhancing understanding and providing alternative proofs for classical functions.

Findings

New necessary and sufficient condition for zeros of derivatives

Reproves zeros of Xi, Gamma, and digamma functions using the new theorem

Provides a unified approach to zeros of derivatives in complex analysis

Abstract

The main result of this paper shows a totally new necessary and sufficient condition to determine both real and complex zeros of derivative of all entire and meromorphic functions of one complex variable in the extended complex plane. By using the theorem, we reprove some results about zeros of derivative of Xi function,Gamma function and digamma function in a new way.