Extending the Geometric Modulus Principle

In a 2011 paper, Kalantari observes that complex polynomials exhibit a certain symmetry with respect to argument at every point. To wit, if the polynomial p(z) has order m as an analytic function whose germ is centered at the point z = z₀, then the complex plane near zo locally splits into 2m sectors of equal angle, alternating between sectors in which |p(z)| > |p(z₀)| and sectors in which |p(z)| < |p(z₀)|. In our subsequent research, we found that this symmetry, which Kalantari formalizes as the Geometric Modulus Principle, is retained (under appropriate modifications) for much larger classes of functions, in particular holomorphic functions as well as harmonic functions of two real variables. This dissertation continues our work in several respects. First, we expand the Geometric Modulus Principle to a broader class of analytic functions, the so-called approximately harmonic functions, that behaves much like the class of harmonic functions with respect to ring operations. We then define the geometric modulus property, enjoyed by a larger-still class of functions, which formalizes the idea of the sign of a function "undulating" in sectors around a point of its domain. If a function has the geometric modulus property and is of order m as an analytic function at a point, then its mᵗʰ derivative with respect to radius is a trigonometric polynomial that behaves like a sine function; we call such trigonometric polynomials sinusoidal, devoting a section of the appendix to their properties. We conclude with some applications of our work and questions for future study.Ph.D.Includes bibliographical reference