On reflection principles supported on a finite set

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Abstract

We investigate the existence of reflection formulas supported on a finite set. It is found that for solutions of the Laplace and Helmholtz equation there are no finitely supported reflection principles unless the support is a single point. This confirms that in order to construct a reflection formula that is not ‘point to point’, it is necessary to consider a continuous support. For solutions of the wave equation 2u/xy=0, there exist finitely supported reflection principles that can be constructed explicitly. For solutions of the telegraph equation 2u/xy+λ2u=0, we show that if a reflection principle is supported on less than five points then it is a point to point reflection principle.

Keywords

Schwarz reflection principle
Reflection formula
Helmholtz equation
Laplace equation
Wave equation
Eigenvalue equation
Telegraph equation

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This project was carried out while the author was a student of the Mathematics Department of the University of South Florida.