Coherence and Quantum Optics V pp 1051-1056 | Cite as

# Complex Variable Representations of Scattered Scalar Wavefields

## Abstract

The representation of scalar wavefields by means of functions of a complex variable has an interpretative value in optics^{l}. 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 Physics^{2}. 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 analytic^{3} or pseudoanalytic functions^{4}. 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 Oseen^{5} for scalar wavefields.

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