Zeros of a Family of Complex-Valued Harmonic Functions with Poles

We study the zeros of a family of complex-valued harmonic functions \(f_c(z)=z^n+c/\overline{z}^k-1\) $f_c(z)=z^n+c/{\overline{z}}^k-1$ for c a non-zero complex number. We use the harmonic analogue of the Argument Principle. The critical curve separating the sense-preserving and sense-reversing regions for \(f_c\) $f_c$ is a circle \(\Gamma _c\) ${\mathrm{\Gamma }}_c$ whose image under \(f_c\) $f_c$ is an epicycloid. Thus, after studying the geometry of these curves, we can determine the winding number of \(f_c(\Gamma _c)\) $f_c({\mathrm{\Gamma }}_c)$ about the origin for all values of c and count the zeros of \(f_c\) $f_c$ in the complex plane.