Applying the argument principle to count zeros of harmonic functions with poles

Several recent papers investigate the way in which the number of zeros of a complex-valued harmonic function depends on its coefficients by analyzing specific simple families. These families share two features: (1) the critical curve separating the sense-preserving and sense-reversing regions is a circle, and (2) the image of that circle is a well-understood parametric curve. In such cases, the harmonic analogue of the Argument Principle can be applied to count the zeros. In this paper, we illustrate the strengths and limitations of these techniques; we construct a new family of complex-valued harmonic functions with poles having some of these features. We obtain detailed zero-counting theorems for two subfamilies, and we illustrate how to obtain less detailed zero-counting theorems for the general family.