On the analytic functions

TL;DR

This paper proves that any integrable function on a smooth, star-shaped boundary with zero integral can be realized as the boundary value of an analytic function inside the domain.

Contribution

It establishes a new boundary value representation result for analytic functions on star-shaped domains with smooth boundaries.

Findings

Any zero-mean integrable boundary function is a boundary value of an analytic function.

The result extends boundary value theory for holomorphic functions in star-shaped domains.

Provides a constructive approach to boundary value problems for analytic functions.

Abstract

Let $\Omega$ be a connected bounded domain on the complex plane, $S$ be its boundary, which is closed, star-shaped, $C^1$-smooth, and $H(\Omega)$ is the set of analytic (holomorphic) in $\Omega$ functions. The aim of this paper is to prove that an arbitrary $f\in L^1(S)$, satisfying the condition $\int_Sf(s)ds=0$, can be boundary value of an $f\in H(\Omega)$.