Accession Number : AD0621238

Title :   INTEGRAL REPRESENTATIONS FOR HARMONIC FUNCTIONS IN THREE REAL VARIABLES.

Descriptive Note : Master's thesis,

Corporate Author : PENNSYLVANIA STATE UNIV UNIVERSITY PARK

Personal Author(s) : Becker,Dorothy Graham

Report Date : JAN 1961

Pagination or Media Count : 62

Abstract : Let X = (x,y,z) be a point in three-dimensional Euclidean space and f(u,v) an analytic function of two complex variables u and v; u,v and x being connected by the relation u = 1/2(iy + z) + x + + 1/2(iy - z). The Whittaker-Bergman operator (1/2 pi-i) integral of f(u,v)dv/v 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 power/(u - iA), A a real constant. For k>0 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 = =1/2, =3, =3 1/2 and for points inside and outside the circle y-sq + z-sq = 1. (Author)

Descriptors :   (*COMPLEX VARIABLES, FUNCTIONS(MATHEMATICS)), (*INTEGRALS, FUNCTIONS(MATHEMATICS)), HARMONIC ANALYSIS, FUNCTIONAL ANALYSIS, OPERATORS(MATHEMATICS), PARTIAL DIFFERENTIAL EQUATIONS

Distribution Statement : APPROVED FOR PUBLIC RELEASE