Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 2H (Topics in Analysis)

Runge approximation theorem in its polynomial form says that if $K\subset \mathbb{C}$ is compact with connected complement and $f$ is holomorphic on a neighborhood of $K$, then polynomials approximate $f$ uniformly on $K$.

True when “uniform” means on all of the unbounded quadrant $\Omega$. A uniformly convergent sequence $(P_n)$ is uniformly Cauchy. For sufficiently large $m,n$, the polynomial $P_n-P_m$ is bounded on $\Omega$. Every nonconstant polynomial is unbounded along some ray contained in $\Omega$, so $P_n-P_m$ is constant. Fixing one large $m$, the limit is therefore $P_m$ plus the limit of constants, hence is a polynomial.

True. Exhaust the quadrant by compact sets $K_n$ with connected complements and with every compact subset of $\Omega$ eventually contained in the interior of $K_n$. Runge's theorem gives a polynomial $P_n$ satisfyingThus $P_n\to f$ locally uniformly. For a fixed $z\in \Omega$, choose a small circle about $z$ eventually contained in every $K_n$. The Cauchy integral formula for derivatives applied to $P_n-f$ givesfor every $r\ge 0$.