Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 2I / Solution

Runge's polynomial approximation theorem says that if $K\subset \mathbb{C}$ is compact, $\mathbb{C}\setminus K$ is connected, and $f$ is holomorphic on a neighborhood of $K$, then for every $\epsilon >0$ there is a polynomial $p$ such that

LetThis arc is compact, its complement is connected, and $1/z$ is holomorphic on a neighborhood of it. Applying Runge with error $1/n$ gives a polynomial $q_n$ satisfyingwhich is the requested uniform approximation.

For the pointwise construction, for $n\ge 2$ setThe three pieces are disjoint compact sets and $\mathbb{C}\setminus K_n$ is connected. Define a function on a neighborhood of $K_n$ to equal $1$, $-1$, and $0$ on neighborhoods of $K_n^{+}$, $K_n^{-}$, and $K_n^0$, respectively. It is holomorphic because those neighborhoods may be chosen disjoint. Runge supplies a polynomial $P_n$ such thatEvery fixed point with $|z|<1$ eventually belongs to the appropriate one of these three sets, according to the sign of its real part. These inequalities therefore give precisely the asserted pointwise limits.

Solved by gpt-5.6-sol high.