# 3.II.13E

(a) State Taylor's Theorem.

(b) Let $f(z)=\sum_{n=0}^{\infty} a_{n}\left(z-z_{0}\right)^{n}$ and $g(z)=\sum_{n=0}^{\infty} b_{n}\left(z-z_{0}\right)^{n}$ be defined whenever $\left|z-z_{0}\right|. Suppose that $z_{k} \rightarrow z_{0}$ as $k \rightarrow \infty$, that no $z_{k}$ equals $z_{0}$ and that $f\left(z_{k}\right)=g\left(z_{k}\right)$ for every $k$. Prove that $a_{n}=b_{n}$ for every $n \geqslant 0$.

(c) Let $D$ be a domain, let $z_{0} \in D$ and let $\left(z_{k}\right)$ be a sequence of points in $D$ that converges to $z_{0}$, but such that no $z_{k}$ equals $z_{0}$. Let $f: D \rightarrow \mathbb{C}$ and $g: D \rightarrow \mathbb{C}$ be analytic functions such that $f\left(z_{k}\right)=g\left(z_{k}\right)$ for every $k$. Prove that $f(z)=g(z)$ for every $z \in D$.

(d) Let $D$ be the domain $\mathbb{C} \backslash\{0\}$. Give an example of an analytic function $f: D \rightarrow \mathbb{C}$ such that $f\left(n^{-1}\right)=0$ for every positive integer $n$ but $f$ is not identically 0 .

(e) Show that any function with the property described in (d) must have an essential singularity at the origin.