Certain properties of functions harmonic within a sphere
Ernest P. Miles · Proceedings of the American Mathematical Society · 1951
Introduction.1 Let 5 be the sphere of radius a about the origin O of the rectangular coordinate system (x, y, z).Let V be the interior of 5. Let U= £/(x, y, z) = ¿7(r, 9, 4>) be the function, harmonic in V, given by the Poisson integral of w(0, c6) over S,where w(0, 4>)EL on 5. When the relation (1.1) holds between {/and u, we write U = piu).The following Theorem I extends to three dimensions results obtained by Douglas for two dimensions (see [2, pp.307-311 ]).2 Theorem I. Let w(P) EL on S and U be the function, harmonic in V, such that U = piu).Let P" (cos 6) and P™(cos 0) be respectively the Legendre polynomial of order n, and the associated Legendre function of the first kind.Denote the Laplace expansion of u on S by U(fl, ) ~ È AnPniC0S 0) n=0 L n + 2~1 (-4n.m cos m + Bn.m sin m )P" (cos 0) m=l Then, if any one of the three numbers A[U]= fff \VUiP)\2dVP, «11 * CC M CC [