INTEGRAL REPRESENTATIONS FOR HARMONIC FUNCTIONS IN THREE REAL VARIABLES.
PENNSYLVANIA STATE UNIV UNIVERSITY PARK
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Let X x,y,z be a point in three-dimensional Euclidean space and fu,v an analytic function of two complex variables u and v u,v and x being connected by the relation u 12iy z x 12iy - z. The Whittaker-Bergman operator 12 pi-i integral of fu,vdvv is known to represent a complex harmonic function H in a neighborhood of some fixed point X sub 0. Bergman has described the function and its singularities in case f v to the Kth poweru - iA, A a real constant. For k0 H is a twovalued harmonic function which for k 0 is singular along the negative x-axis in the first sheet and for A not 0 branches along the circle x 0, y-sq z-sq A-sq. The traces of the level surfaces Re H 0 in the plane x 0 are studied. For A 0 such traces are rose curves of k leaves for k odd and 2k leaves for k even. For A 1 the traces are obtained for C 12, 3, 3 12 and for points inside and outside the circle y-sq z-sq 1. Author