Location of boundary contours and discontinuity arcs, of known shape and conditions, of analytic functions by using contour integrals

摘要

Complex integrals independent of the closed contour C are used for the determination of the position of a boundary contour L and/or of an arc of discontinuity (of an arbitrary shape but known in advance) of an analytic function in the complex plane which can be expressed with the help of an analytic function even on L (not known) and a Cauchy type integral on L (with a density known in advance). This method constitutes a generalization of the methods based on complex integrals independent of the contour C for the determination of the position of the zeros (and/or the poles) of an analytic function Φ(z). The values of Φ(z) (or one of its derivatives) on C only are necessary for the application of the method proposed here. Moreover, the present results are of importance in plane elasticity for the determination of the position of holes and/or inclusions and/or cracks of arbitrary but known shape (instead of the contour L). Many generalizations and applications of the present method are possible.