Difference quotient of holomorphic function is holomorphic: Difference between revisions

Revision as of 20:14, 26 April 2008

Suppose $U \subset C$ is an open subset and $f : U \to C$ is a holomorphic function. Consider the function $F : U \times U \to C$ given by:

$F (z, w) : = \frac{f (z) - f (w)}{z - w}, (z \neq w), f^{'} (z), (z = w)$

Then, for any fixed value of $w \in C$ , the function:

$z \mapsto F (z, w)$

is holomorphic.

@@ Line 3: / Line 3: @@
 Suppose <math>U \subset \mathbb{C}</math> is an open subset and <math>f:U \to \mathbb{C}</math> is a [[holomorphic function]]. Consider the function <math>F: U \times U \to \mathbb{C}</math> given by:
-<math>F(z,w) := \frac{f(z) - f(w)}{z - w}, (</math> if <math>z \ne w),  \qquad f'(z), (</math> if <math>z = w)</math>
+<math>F(z,w) := \frac{f(z) - f(w)}{z - w}, (z \ne w),  \qquad f'(z), (z = w)</math>
 Then, for any fixed value of <math>w \in \mathbb{C}</math>, the function: