Difference quotient of holomorphic function is holomorphic - Revision history

Vipul: 3 revisions

2008-05-18T19:12:04Z

3 revisions

← Older revision	Revision as of 19:12, 18 May 2008
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Vipul at 20:14, 26 April 2008

2008-04-26T20:14:23Z

← Older revision		Revision as of 20:14, 26 April 2008
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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}, (~~</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:		Then, for any fixed value of <math>w \in \mathbb{C}</math>, the function:

Vipul at 20:14, 26 April 2008

2008-04-26T20:14:01Z

← Older revision		Revision as of 20:14, 26 April 2008
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:		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:

Vipul: New page: ==Statement== Suppose $U \subset \mathbb{C}$ is an open subset and $f:U \to \mathbb{C}$ is a holomorphic function. Consider the function $F: U \times U \to ...$

2008-04-26T20:13:29Z

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 ...

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 \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>

Then, for any fixed value of <math>w \in \mathbb{C}</math>, the function:

<math>z \mapsto F(z,w)</math>

is holomorphic.

Difference quotient of holomorphic function is holomorphic - Revision history

Vipul: 3 revisions

Vipul at 20:14, 26 April 2008

Vipul at 20:14, 26 April 2008

Vipul: New page: ==Statement== Suppose U \subset \mathbb{C} is an open subset and f:U \to \mathbb{C} is a holomorphic function. Consider the function F: U \times U \to ...

Vipul: New page: ==Statement== Suppose $U \subset \mathbb{C}$ is an open subset and $f:U \to \mathbb{C}$ is a holomorphic function. Consider the function $F: U \times U \to ...$