Cauchy integral formula for derivatives: Difference between revisions
(New page: ==Statement== Suppose <math>U</math> is a domain in <math>\mathbb{C}</math> and <math>f:U \to \mathbb{C}</math> is a holomorphic function. Suppose <math>\gamma</math> is the circl...) |
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==Statement== | ==Statement== | ||
Suppose <math>U</math> is a [[domain]] in <math>\mathbb{C}</math> and <math>f:U \to \mathbb{C}</math> is a [[holomorphic function]]. Suppose <math>\gamma</math> is the circle of radius <math>r</math> centered at a point <math>z_0 \in U</math>, such that <math>\gamma</math> lies completely inside <math>U</math>. Then, we have: | Suppose <math>U</math> is a [[domain]] in <math>\mathbb{C}</math> and <math>f:U \to \mathbb{C}</math> is a [[holomorphic function]]. Suppose <math>\gamma</math> is the circle of radius <math>r</math> centered at a point <math>z_0 \in U</math>, such that <math>\gamma</math> lies completely inside <math>U</math>. Then, for any point <math>z \in U</math>, we have: | ||
<math>f^{(n)}( | <math>f^{(n)}(z) = \frac{n!}{2\pi i} \oint_\gamma \frac{f(\xi)}{(\xi - z)^{n+1}} \, d\xi</math> | ||
==Applications== | |||
* [[Cauchy estimates for derivatives]] | |||
Latest revision as of 19:10, 18 May 2008
This article gives the statement, and possibly proof, of a basic fact in complex analysis.
View a complete list of basic facts in complex analysis
Statement
Suppose is a domain in and is a holomorphic function. Suppose is the circle of radius centered at a point , such that lies completely inside . Then, for any point , we have:
