On the domain of regularity of generalized axially symmetric potentials

which depends analytically on the two real variables x and y in a domain containing a segment of the singular line y =0, is uniquelydetermined by its values on y =0. In this note we shall consider the following problem: If the function g(z) = u(z, 0) is continued to complex values of z, to what extent does its domain of regularity dete, mine the domain of regularity of u(x, y)? A partial answer to this has been given by Erdelyi [1], who proved that if g(z) is holomorphic in a y-convex domain2 p, the function u(x, y) is regular at all real points (x, y) for which x+iyEEp. The general results of I. N. Vekua in the theory of elliptic differential equations with analytic coefficients' suggest the following wider (and perhaps more natural) statement:

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