THESIS
2006
ix, 28 leaves : ill. ; 30 cm
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:
^{2} ^{3}...[
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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)+i
^{2}f(z+ε+εi)+i
^{3}f(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.
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