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Location of Zeros of Holomorphic Functions

Leonardo de Lima

Abstract

In this article, various results will be demonstrated that enable the delimitation of a zero-free region for holomorphic functions on a set $K$, studying the behavior of their imaginary or real part on the boundary of $K$. These findings contribute to a deeper understanding of the distribution of zeros, shedding light on the intricate nature of holomorphic functions within the specified set.

Location of Zeros of Holomorphic Functions

Abstract

In this article, various results will be demonstrated that enable the delimitation of a zero-free region for holomorphic functions on a set , studying the behavior of their imaginary or real part on the boundary of . These findings contribute to a deeper understanding of the distribution of zeros, shedding light on the intricate nature of holomorphic functions within the specified set.
Paper Structure (6 sections, 10 theorems, 25 equations, 2 figures)

This paper contains 6 sections, 10 theorems, 25 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Harmonic Functions
  3. Delimitation of Zero-Free Regions for Holomorphic Functions
  4. Application
  5. Gamma function
  6. Zeta function

Key Result

Theorem 2.1

Let $u: U \to \mathbb{R}$ be a harmonic function. If $K$ is a non-empty compact subset of $U$, then $u$ restricted to $K$ attains its maximum and minimum on the boundary of $K$. If $U$ is connected, this means that $u$ cannot have local maxima or minima, except in the case where $u$ is constant.

Figures (2)

  • Figure 1: Graph of the real and imaginary parts of $F(\sigma+10i)$.
  • Figure 2: Graph of the real and imaginary parts of $F(\sigma+10i)$.

Theorems & Definitions (21)

  • Theorem 2.1: Maximum Principle
  • proof
  • Theorem 2.2
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.1
  • proof
  • Corollary 3.1
  • proof
  • ...and 11 more