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2008-05-18T19:11:56Z

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Vipul: New page: ==Definition== Suppoes $U \subset \mathbb{C}$ is an open subset and $f:U \to \mathbb{C}$ is a function. The '''difference quotient''' of $f$ is a function...

2008-04-26T20:10:05Z

New page: ==Definition== Suppoes <math>U \subset \mathbb{C}</math> is an open subset and <math>f:U \to \mathbb{C}</math> is a function. The '''difference quotient''' of <math>f</math> is a function...

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==Definition==

Suppoes <math>U \subset \mathbb{C}</math> is an open subset and <math>f:U \to \mathbb{C}</math> is a function. The '''difference quotient''' of <math>f</math> is a function <math>F: (U \times U) \setminus (diag) \to \mathbb{C}</math> given by:

<math>F(z,w) := \frac{f(z) - f(w)}{z - w}</math>

If <math>f</math> is a [[holomorphic function]], then <math>F</math> can be extended to a continuous function on <math>U \times U</math>, by defining:

<math>F(z,z) := f'(z)</math>

==Facts==

* [[Difference quotient of holomorphic function is holomorphic]]

Difference quotient of a complex-valued function - Revision history

Vipul: 1 revision

Vipul: New page: ==Definition== Suppoes U \subset \mathbb{C} is an open subset and f:U \to \mathbb{C} is a function. The '''difference quotient''' of f is a function...

Vipul: New page: ==Definition== Suppoes $U \subset \mathbb{C}$ is an open subset and $f:U \to \mathbb{C}$ is a function. The '''difference quotient''' of $f$ is a function...