There is a proof of [Mittag-Leffler's theorem](http://en.wikipedia.org/wiki/Mittag-Leffler_theorem) with an explicit construction of a holomorphic function with the prescribed poles with prescribed order and residues, for a countable discrete set of points. I do not remember the reference; but my memory from my graduate course is that one defines a series sum and make certain adjustments. I was never quite good in this type of processes; so I am facing problem with the following exercise, which is nagging me for a long time. I thought of using Math Overflow with the hope that somebody can help me out.


Now I want to prove that every open set in the complex plane is now a [domain of holomorphy](http://en.wikipedia.org/wiki/Mittag-Leffler_theorem). We take the boundary $\partial \Omega$ of the open set $\Omega$, and we take a countable dense sequence of points $z\_i$ in $\partial \Omega$. If we are able to construct a series sum with poles at $z\_i$, but so that it converges absolutely and uniformly on every compact set in the interior of $\Omega$, then we are done.


I would be most grateful if somebody can show me how to do the above.



Domains of holomorphy in the complex plane
Let $\zeta\_k$ be a countable dense sequence of points in the boundary and consider $f(z) = \sum \frac{1}{2^k} \frac{1}{z-\zeta\_k}$. The sum is plainly uniformly convergent on any subset of finite distance from the boundary, in particular on any compact subset of the interior. 


