Treffer: Complex‐distance potential theory, wave equations, and physical wavelets: Complex-distance potential theory, wave equations, and physical wavelets

Potential theory in ℝn is extended to ℂn by analytically continuing the Euclidean distance function. The extended Newtonian potential ϕ(z) is generated by a (non‐holomorphic) source distribution δ˜(z) extending the usual point source δ(x). With Minkowski space ℝn, 1 embedded in ℂn+1, the Laplacian Δn+1 restricts to the wave operator □n,1 in ℝn, 1. We show that δ˜(z) acts as a propagator generating solutions of the wave equation from their initial values, where the Cauchy data need not be assumed analytic. This generalizes an old result by Garabedian, who established a connection between solutions of the boundary‐value problem for Δn+1 and the initial‐value problem for □n,1 provided the boundary data extends holomorphically to the initial data. We relate these results to the physical avelets introduced previously. In the context of Clifford analysis, our methods can be used to extend the Borel–Pompeiu formula from ℝn+1 to ℂn+1, where its riction to Minkowski space ℝn, 1 provides solutions for time‐dependent Maxwell and Dirac equations. Copyright © 2002 John Wiley & Sons, Ltd.