ABSTRACT.

Let \( D \) be a connected bounded domain in \( \mathbb{R}^{2} \), \( S \) be its boundary which is closed, connected and smooth or \( S = (-\infty,\infty) \). Let

\[
\Phi(z) = \frac{1}{2\pi i}\int_{S} \frac{f(s)\, ds}{s – z},
\qquad f \in L^{1}(S), \; z = x + iy.
\]

The singular integral operator

\[
Af := \frac{1}{\pi i}\int_{S} \frac{f(s)\, ds}{s – t},
\qquad t \in S,
\]

is defined in a new way. This definition simplifies the proof of the existence of \( \Phi(t) \). Necessary and sufficient conditions are given for \( f \in L^{1}(S) \) to be a boundary value of an analytic function in \( D \). The Sokhotsky–Plemelj formulas are derived for \( f \in L^{1}(S) \). Our new definition allows one to treat singular boundary values of analytic functions.