Diffraction by a quarter-plane. Analytical continuation of spectral functions

TL;DR

This paper develops a novel method for analytically continuing spectral functions in the diffraction problem by a quarter-plane, using integral formulas and a new concept called additive crossing of branch lines.

Contribution

It introduces new integral formulas and the concept of additive crossing to analytically continue spectral functions in diffraction problems, advancing the mathematical understanding of such issues.

Findings

Derived integral formulas for spectral function continuation

Introduced the concept of additive crossing of branch lines

Reformulated Wiener-Hopf problem using new concepts

Abstract

The problem of diffraction by a Dirichlet quarter-plane (a flat cone) in a 3D space is studied. The Wiener-Hopf equation for this case is derived and involves two unknown (spectral) functions depending on two complex variables. The aim of the present work is to build an analytical continuation of these functions onto a well-described Riemann manifold and to study their behaviour and singularities on this manifold. In order to do so, integral formulae for analytical continuation of the spectral functions are derived and used. It is shown that the Wiener-Hopf problem can be reformulated using the concept of additive crossing of branch lines introduced in the paper. Both the integral formulae and the additive crossing reformulation are novel and represent the main results of this work.