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 z0 interior to a simple closed contour C situated in D in terms of the values of 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 |z-a| = 1/r for which the formula is valid.
Highlights
We make reference to [1] for elementary knowledge in complex analysis used below
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 z − a =1 r for which the formula is valid
Integrals on unbounded contours have been used frequently in complex analysis, they have never appeared in the context of Cauchy integral formula
Summary
We make reference to [1] for elementary knowledge in complex analysis used below. It is known (see [2]) that for every rational function R ( z) of degree n the complex plane can be partitioned into n sets whose interior are fundamental domains of R ( z) , i.e. they are mapped conformally ( bijectively) by R ( z) onto the whole complex plane with some slits.
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