Boundary values of analytic functions

TL;DR

This paper investigates the boundary behavior of certain integral functions related to analytic functions in bounded domains, providing new definitions, necessary and sufficient conditions, and deriving classical formulas for functions in L^1 spaces.

Contribution

It introduces a novel approach to defining boundary values of integral functions and establishes criteria for boundary value functions to be associated with analytic functions in the domain.

Findings

Derived Sokhotsky-Plemelj formulas for L^1 boundary functions.

Provided necessary and sufficient conditions for boundary values of analytic functions.

Introduced a new method for defining boundary values on smooth boundaries.

Abstract

Let $D$ be a connected bounded domain in $\R^2$, $S$ be its boundary which is closed, connected and smooth. Let $\Phi(z)=\frac 1 {2\pi i}\int_S\frac{f(s)ds}{s-z}$, $f\in L^1(S)$, $z=x+iy$. Boundary values of $\Phi(z)$ on $S$ are studied. The function $\Phi(t)$, $t\in S$, is defined in a new way. Necessary and sufficient conditions are given for $f\in L^1(S)$ to be boundary value of an analytic in $D$ function. The Sokhotsky-Plemelj formulas are derived for $f\in L^1(S)$.