## Theses and Dissertations

#### Abstract

The Fundamental Theorem of Algebra is a useful tool in determining the number of zeros of complex-valued polynomials and rational functions. It does not, however, apply to complex-valued harmonic polynomials and rational functions generally. In this thesis, we determine behaviors of the family of complex-valued harmonic functions $f_{c}(z) = z^{n} + \frac{c}{\overline{z}^{k}} - 1$ that defy intuition for analytic polynomials. We first determine the sum of the orders of zeros by using the harmonic analogue of Rouch\'e's Theorem. We then determine useful geometry of the critical curve and its image in order to count winding numbers by applying the harmonic analogue of the Argument Principle. Combining these results, we fully determine the number of zeros of $f_{c}$ for $c > 0$.

MS

#### College and Department

Physical and Mathematical Sciences; Mathematics

2022-12-12

Thesis

#### Handle

http://hdl.lib.byu.edu/1877/etd12650

#### Keywords

complex analysis, complex-valued harmonic function, epicycloid

english

COinS