Cauchy-Type Integral

For functions of a complex variable there is a known Cauchy integral theorem which is formulated as follows. Let L be a smooth closed curve (contour). Hereinafter the term smooth curve (contour) covers a simple (i.e., having no points of self-intersection) closed or open line with a slip tangent and with no cusps. If the function f(ζ) is analytical in the domain D bounded by the contour L and continuous on the contour itself (i.e., continuous in the closed domain D̄ = D∪L), we deal with the following Cauchy formula: 1.1.1$$\frac{1}{{2\pi }}\int\limits_{L} {\frac{{f(\zeta )}}{{\zeta - \zeta '}}d\zeta = } {\text{ }}\left\{ {\begin{array}{*{20}{c}} {f\left( {\zeta '} \right), \zeta ' \in D,} \hfill \\ {0, \zeta ' \in C\bar{D},} \hfill \\ \end{array} } \right. $$ where CD̄ is the complement of the closed domain D̄ with respect to the total complex plane. (From now on it is assumed that the contour is traversed in the positive sense for which the inner domain is always on the left.)