complex functions which can be approximated by sequence of polynomials For which among the following functions $f(z)$ defined on $G=\mathbb{C}\setminus\{0\}$ is there no sequence of polynomials approximating $f(z)$ uniformly on compact subsets of $G$ ?
1)$e^z$
2)$\frac{1}{z}$
3)$z^2$
4)$\frac{1}{z^2}$
here $e^z$ and $z^2$ are entire functions so can I say they can be approximated by sequence of polynomials ?
I know Runge's theorem but not understanding how to apply it.
 A: Let $n$ be a positive integer and let us assume that $1/z^n$ can be approximated uniformly on the compact set $K:=\{z: r\leq |z|\leq 1\}$ with $0<r<1$, by polynomials. Then there is a polynomial $P$ such that for all $z\in K$,
$$|1/z^n - P(z)|\leq 1/2\implies |1-z^nP(z)|\leq |z|^n/2\leq 1/2.$$
Now, since  $1-z^nP(z)$ is a polynomial, by the maximum principle,
$$|1-z^nP(z)|\leq 1/2$$
for every $z$ in the disk $|z|\leq 1$. But $|1-0^n\cdot P(0)|=1>1/2$ and we have a contradiction.
P.S. Runge's theorem guarantees the existence of a sequence of approximating polynomials on a compact $K$ if $\mathbb{C}\setminus K$ is connected. Note that for $K=\{z: r\leq |z|\leq 1\}$ as above, $\mathbb{C}\setminus K$ is NOT connected.
A: Suppose $|f(z)-P(z)|<\epsilon$ (with $P$ a polynomial) on the unit circle $S^1$. If $f(z)=\sum_{n\in {\Bbb Z}} c_n z^n$ is the Laurent expansion of $f$ at zero (converging uniformly on compact subsets of $G$) then by Cauchy and the fact that $P$ is entire we have for $n<0$:
$$ |c_n| = |\oint_{S^1} z^{-n-1} f(z) \frac{dz}{2\pi i}|
  = |\oint_{S^1} z^{-n-1} (f(z)-P(z)) \frac{dz}{2\pi i}| \leq \epsilon$$
As $\epsilon>0$ is arbitrary, $c_n=0$ for every $n<0$ and $f$ must be entire.
