Numerical Calculations in Clifford Algebra

24Cauchy Extension

Cauchy (1841) discovered that an analytic function of a complex variable in the two dimensions of the complex plane can be reproduced everywhere inside a closed boundary from the values of the function on the boundary. Under the right conditions, the extension can be made either to the inner region or the outer region.

The same kind of extension is possible for electromagnetic fields in regions without sources and with uniform and constant material properties when represented in Clifford algebra (McIntosh 1989). The mathematical functions representing the electromagnetic fields are in this case termed monogenic rather than analytic. The monogenic functions behave as a generalisation of complex analytic functions to dimensions higher than two.

24.1 Background

Solutions to problems involving boundary element methods (Brebbia, Telles & Wrobel 1984) yield, in the first instance, field values on some boundary. Often, it is also of interest to evaluate the field away from the boundary, as in the case of the electromagnetic field around an antenna or inside a microwave cavity.

Normally, there is an explicit integral relationship between the field values on the boundary and the field value at any chosen point away from the boundary. The integral usually has one factor, such as a Green's function (Shahpari & Seagar 2020) or a Cauchy kernel, which contains a difference of two terms in the denominator. As the point at which the field is evaluated approaches the boundary, ...