On the [Epsilon]-discrete holomorphic deformations of holomorphic functions

Abstract

In this thesis, we study a special type of functions called ε-discrete holomorphic functions, which are complex-valued functions that satisfy a difference equation on the complex plane for ε > 0: f(z)+if(z+ε)+i2f(z+ε+εi)+i3f(z+εi)=0. This equation can be viewed as a discrete analog of the Cauchy-Riemann equa-tions for ordinary holomorphic functions. We are interested in the real ana-lytic deformations of holomorphic functions into ε-discrete holomorphic func-tions, which are complex-valued functions F(z, t) on C x [0, M), where M > 0, such that for every given t = ε ∈ (0, M) , the one variable function F(z, ε) is ε-discrete holomorphic. We prove that polynomials and exponential functions can be deformed into ε-discrete holomorphic functions, and that these deformations can be expressed as power series. Using these results, we study how holomorphic functions with bounded derivatives at z = 0 can be deformed into ε-discrete holomorphic func-tions which are real analytic. We also study the non-analytic deformations of an arbitrary holomorphic function.

Date of Award	2006
Original language	English
Awarding Institution	The Hong Kong University of Science and Technology