Difference quotient of holomorphic function is holomorphic: Difference between revisions
(New page: ==Statement== 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 ...) |
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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: | 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}, (z \ne w), \qquad f'(z), (z = w)</math> | <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> | ||
Then, for any fixed value of <math>w \in \mathbb{C}</math>, the function: | Then, for any fixed value of <math>w \in \mathbb{C}</math>, the function: | ||
Revision as of 20:14, 26 April 2008
Statement
Suppose is an open subset and is a holomorphic function. Consider the function given by:
if if
Then, for any fixed value of , the function:
is holomorphic.