Properties of Analytic Mappings

Abstract

We now will concentrate on the aspect of analytic functions as mappings from domains in ℝ² into ℝ². For an arbitrary C¹ mapping the relation

$$ f\left( {a + z} \right) - f(a) = \frac{{\partial f}}{{\partial z}}{\left| {z + \frac{{\partial f}}{{\partial \overline z }}} \right|_a}\overline z + o\left( {\left| z \right|} \right) $$

means that the derivative Df|_a of f at a is the real linear mapping

$$ z \to \frac{{\partial f}}{{\partial z}}{\left| {_az + \frac{{\partial f}}{{\partial \overline z }}} \right|_a}\overline z $$

(1.1)

.