The Extension of Cauchy Integral Formula to the Boundaries of Fundamental Domains

Abstract

The Cauchy integral formula expresses the value of a function () f z , which is analytic in a simply connected domain D, at any point 0 z interior to a simple closed contour C situated in D in terms of the values of () 0 f z z z − on C. We deal in this paper with the question whether C can be the boundary ∂Ω of a fundamental domain Ω of () f z. At the first look the answer appears to be negative since ∂Ω contains singular points of the function and it can be unbounded. However, the extension of Cauchy integral formula to some of these unbounded curves, respectively arcs ending in singular points of () f z is possible due to the fact that they can be obtained at the limit as r → ∞ of some bounded curves contained in the pre-image of the circle z r = and of some circles 1 z a r − = for which the formula is valid.