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29 March 2024
 
  » arxiv » math-ph/9908031

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Complex-Distance Potential Theory and Hyperbolic Equations
Gerald Kaiser ;
Date 31 Aug 1999
Journal Clifford Analysis, J. Ryan and W. Sprossig, eds., Birkhauser Progress in Physics, Vol. 19, 2000.
Subject Mathematical Physics; Analysis of PDEs MSC-class: 31-XX, 32-XX, 35-XX, 78-XX | math-ph math.AP math.MP
AbstractAn extension of potential theory in R^n is obtained by continuing the Euclidean distance function holomorphically to C^n. The resulting Newtonian potential is generated by an extended source distribution D(z) in C^n whose restriction to R^n is the delta function. This provides a natural model for extended particles in physics. In C^n, interpreted as complex spacetime, D(z) acts as a propagator generating solutions of the wave equation from their initial values. This gives a new connection between elliptic and hyperbolic equations that does not assume analyticity of the Cauchy data. Generalized to Clifford analysis, it induces a similar connection between solutions of elliptic and hyperbolic Dirac equations. There is a natural application to the time-dependent, inhomogeneous Dirac and Maxwell equations, and the `electromagnetic wavelets’ introduced previously are an example.
Source arXiv, math-ph/9908031
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