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Complex‐distance potential theory, wave equations, and physical wavelets
Author(s) -
Kaiser Gerald
Publication year - 2002
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.390
Subject(s) - mathematics , minkowski space , laplace operator , context (archaeology) , wave equation , potential theory , d'alembert operator , euclidean space , holomorphic function , mathematical analysis , initial value problem , operator (biology) , cauchy distribution , boundary value problem , newtonian potential , propagator , mathematical physics , quantum mechanics , physics , paleontology , biochemistry , chemistry , repressor , gene , transcription factor , gravitation , biology
Potential theory in ℝ n is extended to ℂ n 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 ℝ n , 1 embedded in ℂ n +1 , the Laplacian Δ n +1 restricts to the wave operator □ n ,1 in ℝ n , 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 Δ n +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 ℝ n +1 to ℂ n +1 , where its riction to Minkowski space ℝ n , 1 provides solutions for time‐dependent Maxwell and Dirac equations. Copyright © 2002 John Wiley & Sons, Ltd.