Mittag-Leffler's theorem

Remarks

For $U = \C$ the proof is explicit. List $A$ as $a_1, a_2, \dots$ with $\abs{a_n}$ nondecreasing, and subtract from $p_{a_n}$ a Taylor polynomial $q_n$ at $0$ (empty when $a_n = 0$) so that $\abs{p_{a_n} - q_n} \le 2^{-n}$ on the disc $\abs{z} \le \abs{a_n}/2$. Then $\sum_n (p_{a_n} - q_n)$ converges uniformly on compact sets away from $A$ and gives $f$. For a general open set, Runge's theorem on rational approximation replaces the Taylor polynomials. A classical example is $$\frac{\pi^2}{\sin^2(\pi z)} = \sum_{n \in \Z} \frac{1}{(z - n)^2},$$ where here the plain sum already converges.

Going deeper

For a student

The function $f$ is not unique: one can add any holomorphic function on $U$. The theorem gives partial fraction expansions of meromorphic functions, such as $\pi \cot(\pi z) = 1/z + \sum_{n \ne 0} \left(1/(z-n) + 1/n\right)$. Combined with Weierstrass products, it shows that one can prescribe both the principal parts at $A$ and finitely many Taylor coefficients at other points. In the language of sheaf cohomology, it says that the first Cousin problem is solvable on every open subset of $\C$; see also Runge's theorem.