BOUNDARY VALUES OF ANALYTIC FUNCTIONS

Abstract

Let $D$ be a connected bounded domain in $\mathbb{R}^2, S$ be its boundary which is closed, connected and smooth. Let $\Phi(z)=\frac{1}{2 \pi i} \int_S \frac{f(s) d s}{s-z}$, $f \in L^1(S), \quad z=x+i y$. Then 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 function in $D$. The Sokhotski-Plemelj formulas are derived for $f \in L^1(S)$.

Received: March 20, 2023
Accepted: May 11, 2023