#### 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 $c > 0$.

#### Degree

MS

#### College and Department

Physical and Mathematical Sciences; Mathematics

#### Rights

https://lib.byu.edu/about/copyright/

#### BYU ScholarsArchive Citation

Lee, Alexander, "Zeros of a Family of Complex-Valued Harmonic Rational Functions" (2022). *Theses and Dissertations*. 9812.

https://scholarsarchive.byu.edu/etd/9812

#### Date Submitted

2022-12-12

#### Document Type

Thesis

#### Handle

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

#### Keywords

complex analysis, complex-valued harmonic function, epicycloid

#### Language

english