Abstract:
Let $\Gamma $ be a simple closed Lyapunov contour with finite complex measure $\nu$, and let $G^+ $ be the bounded and $G^- $ the unbounded domains with boundary $\Gamma$. Using new notions (so-called $N$-integration and $N^+$- and $N^-$-integrals), we prove that the Cauchy-type integrals $F^+(z)$, $z\in G^+$, and $F^-(z)$, $z\in G^-$, of $\nu $ are Cauchy $N^+$- and $N^-$-integrals, respectively. In the proof of the corresponding results, the additivity property and the validity of the change-of-variable formula for the $N^+$- and $N^-$-integrals play an essential role.
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