A note on analyticity properties of far field patterns

TL;DR

This paper proves that the far field pattern in scattering theory is jointly real analytic in both the incident wave direction and observation direction, enhancing understanding of wave scattering properties.

Contribution

The paper establishes the joint real analyticity of the far field pattern, a property previously known only for separate variables, providing new insights into scattering phenomena.

Findings

Far field pattern is jointly real analytic in both variables

Enhances theoretical understanding of wave scattering

Potential implications for inverse scattering problems

Abstract

In scattering theory the far field pattern describes the directional dependence of a time-harmonic wave scattered by an obstacle or inhomogeneous medium, when observed sufficiently far away from these objects. Considering plane wave excitations, the far field pattern can be written as a function of two variables, namely the direction of propagation of the incident plane wave and the observation direction, and it is well-known to be separately real analytic with respect to each of them. We show that the far field pattern is in fact a jointly real analytic function of these two variables.