Let \(f: U \rightarrow \mathbb{C}\) be analytic and let \(\{|z - z_0| < r \subset U\}\). Then in this disk, \(f\) has a power series representation:

\[ \begin{aligned} f(z) = \sum_{k=0}^\infty a_k (z - z_0)^k, \quad |z - z_0| < r, \quad \text{where } a_k = \frac{f^{(k)}(z_0)}{k!} \\ \end{aligned} \]

The radius of convergence of this power series is \(R \ge r\). Note this means an analytic function is determined entirely in a disk by all it's derivatives \(f^{(k)}(z_0)\) at the center \(z_0\) of the disk!

In other words, if \(f\) and \(g\) are analytic in \(\{|z - z_0| < r\}\) and if \(f^{(k)}(z_0) = g^{(k)}(z_0)\) for all k, then \(f(z) = g(z)\) for all \(z\) in \(\{|z - z_0| < r\}\) !

More at Analysis of a Complex Kind.

# posted by rot13(Unafba Pune) @ 4:34 PM