Boundary Values of Potential Functions Prescribed Along Interior Paths

Arthur S Peters

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1). The integral in (3. 1) can be written i---a^ f tj>(cQ da 27r ^ [a-re^^Ha-re 1 ^] i9 or, if we replace e by z, as o o 2tt r -a P ft (a) da 2Tr - [«. R«- ao «][a. J2S] This integral defines a function of the complex variable z = pe i6, analytic in the domain a/r < |z| < r/a. It shows that (3. 1) possesses a solution only if the prescribed function f(9) is an analytic periodic function of 9 which allows continua- tion into the annular domain defined above. If this condition is satisfied, the equation (3. 1) can presumably be solved by using any - ■ : 17 using any one of the methods of Section 2. [In order to solve (5. 1) by the transform method III, this method must of course be changed to a finite transform method. ] Let us choose method II v;hich depends on the existence of a semi-group property of the transform because of the four methods this one seems the most challenging, and most likely to involve a nev; Idea. If we put A : a/r < 1 vie can virite (J-. L) In the form (j)(q) da (3.

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