In this thesis we study complex harmonic functions of the form f where f is the sum of a nonconstant analytic and a nonconstant anti-analytic function of one variable. The Fundamental Theorem of Algebra does not apply to such functions, so we ask how many zeros a complex harmonic function can have and where those zeros are located. This thesis focuses on the complex harmonic family of polynomials p_c where p_c is the sum of z+(c/2)z^2 and the conjugate of (c/(n-1))z^(n-1)+(1/n)z^n. We first establish properties of the critical curve, which separates orientation preserving and reversing regions. These properties are then used to show the sum of the orders of the zeros of p_c is -n. In turn, we use this to show p_c has n+2 zeros when 04 and n+4 zeros when c>4, n>5. The total number of zeros of p_c changes when zeros interact with the critical curve, so we investigate where zeros occur on the critical curve to understand how the number of zeros of p_c changes for c between 1 and 4.
College and Department
Physical and Mathematical Sciences; Mathematics
BYU ScholarsArchive Citation
Sandberg, Samantha, "Zeros of a Family of Complex Harmonic Polynomials" (2021). Theses and Dissertations. 9554.
complex analysis, harmonic polynomials