Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 3 / 2F (Topics in Analysis)

The polynomial form of Runge theorem says that a function holomorphic on a neighbourhood of a compact set $K\subset \mathbb{C}$ with connected complement can be approximated uniformly on $K$ by polynomials.

For $n\ge 3$, takeThe complement is connected: the missing angular sector connects the inner disk to the exterior, and removing the single point zero from the inner disk does not disconnect it. On a neighbourhood of the annular sector choose an analytic logarithm with arguments between $-1/(2n)$ and $2\pi -1/(2n)$. Its exponential $\exp (3\log z/2)$ agrees with the prescribed function on the sector. On a disjoint small neighbourhood of zero prescribe the analytic function zero. These two neighbourhoods define a holomorphic function $g_n$ near $K_n$ with $g_n=f$ on $K_n$.

By Runge theorem, choose a polynomial $p_n$ with ${\sup }_{K_n}|p_n-g_n|<1/n$. Every fixed nonzero point of the disk eventually belongs to all $K_n$, including points on the positive real axis; zero belongs to every $K_n$. ConsequentlyEach polynomial restricts to an analytic function on $D$. The excluded sector lies immediately below the positive axis and shrinks with $n$, so the construction correctly preserves the chosen value on the branch cut. It gives pointwise convergence, not local uniform convergence across that cut. This is pointwise approximation by a Runge exhaustion.