Cauchy integral formula for derivatives: Difference between revisions

Revision as of 13:50, 19 April 2008

Statement

Suppose $U$ is a domain in $\mathbb {C}$ and $f:U\to \mathbb {C}$ is a holomorphic function. Suppose $\gamma$ is the circle of radius $r$ centered at a point $z_{0}\in U$ , such that $\gamma$ lies completely inside $U$ . Then, for any point $z\in U$ , we have:

$f^{(n)}(z)={\frac {n!}{2\pi i}}\oint _{\gamma }{\frac {f(\xi )}{(\xi -z)^{n+1}}}\,d\xi$

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 ==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)}(z_0) = \frac{n!}{2\pi i} \oint_\gamma \frac{f(z)}{(z - z_0)^{n+1}} \, dz</math>
+<math>f^{(n)}(z) = \frac{n!}{2\pi i} \oint_\gamma \frac{f(\xi)}{(\xi - z)^{n+1}} \, d\xi</math>