Holomorphic function: Difference between revisions

Revision as of 19:29, 13 April 2008

Definition

Definition with symbols

Let $\Omega$ be an open subset of $\mathbb {C}$ . A function $f:U\to \mathbb {C}$ is termed a holomorphic function if it satisfies the following equivalent conditions:

$f$ is complex-differentiable at every point of $U$
$f$ is complex-differentiable at every point of $U$ , and the function $f':U\to \mathbb {C}$ we obtain as the derivative, is a continuous function.
$f$ is infinitely often complex-differentiable. In other words, we can take the $n^{th}$ derivative of $f$ for any $n$
For any point $z_{0}\in U$ , and any disc centered at $z$ of radius $r$ , that lies completely inside $U$ , $f$ can be expressed using a power series in $(z-z_{0})$

Equivalence of definitions

Definitions (1) and (2) are equivalent as a consequence of Morera's theorem.

@@ Line 6: / Line 6: @@
 # <math>f</math> is [[complex-differentiable function|complex-differentiable]] at every point of <math>U</math>
-# <math>f</math> is [[complex-differentiable function|complex-differentiable]] at every point of <math>U</math>, and the function <math>f':U \to mathbb{C}</math> we obtain as the derivative, is a continuous function.
+# <math>f</math> is [[complex-differentiable function|complex-differentiable]] at every point of <math>U</math>, and the function <math>f':U \to \mathbb{C}</math> we obtain as the derivative, is a continuous function.
 # <math>f</math> is infinitely often complex-differentiable. In other words, we can take the <math>n^{th}</math> derivative of <math>f</math> for any <math>n</math>
 # For any point <math>z_0 \in U</math>, and any disc centered at <math>z</math> of radius <math>r</math>, that lies completely inside <math>U</math>, <math>f</math> can be expressed using a power series in <math>(z - z_0)</math>