Abstract

A planar harmonic mapping is a complex-valued function ƒ : D → C of the form ƒ(x+iy) = u(x,y) + iv(x,y), where u and v are both real harmonic. Such a function can be written as ƒ = h+g where h and g are both analytic; the function w = g'/h' is called the dilatation of ƒ. This thesis considers the convolution or Hadamard product of planar harmonic mappings that are the vertical shears of the canonical half-plane mapping p;(z) = z/(1-z) with respective dilatations e^iθz and e^ipz, θ, p ∈ R. We prove that any such convolution is univalent. We also derive a convolution identity that extends this result to shears of p(z) = z/(1-z) in other directions.

Degree

MS

College and Department

Physical and Mathematical Sciences; Mathematics

Rights

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

Date Submitted

2013-07-05

Document Type

Thesis

Handle

http://hdl.lib.byu.edu/1877/etd6425

Keywords

harmonic mapping, shearing, convolution, univalent, dilatation

Included in

Mathematics Commons

Share

COinS