The Hilbert Problem with Constant Coefficient

Consider a sectionally smooth curve L in the complex plane, in the sense defined in Sect. A2.2. We introduce the concept of a sectionally analytic function F(z), with respect to L. This is a function analytic on the complex plane, excluding L and possibly infinite points. It has definite limits as z approaches a point on L from either side (in general not equal), with the exception of certain isolated points which we take to be end points of individual arcs included in L. Near such an isolated point, c, a sectionally analytic function, must obey the condition 

Such points will be termed exceptional points. 

The Hilbert problem with constant coefficient may now be posed. The discussion is based on that of Muskhelishvili (1%3). Given L, we seek a sectionally analytic function F(z) such that its limits at a point u on L, from, S , namely (m), F (m) obey the relation 

The possible exclusion of end points from (A2.3.2) is related to the different and generally singular behaviour of Cauchy integrals at the end points of arcs. 

Let us consider first the simple special case where / = 1. In this case. 

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