by dipali mali
Last Updated July 17, 2017 10:20 AM

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.

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.

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