Complex Variable Representations of Scattered Scalar Wavefields
The representation of scalar wavefields by means of functions of a complex variable has an interpretative value in opticsl. It leads to the underlaying concept of analyticity as a fundamental characteristic. The analytic continuation of solutions to partial differential equations has been discussed in several contexts in Physics2. Like in the case of some elliptic equations, solutions of the two-dimensional Helmholtz equation may be studied in the context of complex variable theory. It will be shown that functions of a complex variable, associated to two-dimensional scalar wavefields, are generalizations of analytic functions, resembling a similarity with generalized analytic3 or pseudoanalytic functions4. There exists a generalized Cauchy integral formula for those functions, which yields a Poisson representation and the integral theorem of Helmholtz and Kirchhoff. This Cauchy integral also provides a framework for the interpretation of inverse diffraction problems and other subjects such as the extinction theorem of Ewald and Oseen5 for scalar wavefields.
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