= Solution
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 $n\geq3$, take
$$
K_n=\{0\}\cup\{z:1/n\leq|z|\leq1-1/n,\ 0\leq\arg z\leq2\pi-1/n\}.
$$
The 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 $K_n$. Consequently
$$
\boxed{p_n(z)\longrightarrow f(z)\quad\text{for every }z\in D.}
$$
Each <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>.
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