Morera's Theorem and annuli I'm trying to get a better understanding of Morera's Theorem for my exam next week. The main application I'm interested in is analyticity. An example problem my professor did in class was: if $G$ is a region and $g_n\to g$ uniformly on $G$, then $g$ is analytic. If I take any triangular path $T$ in $G$ so that the winding number is $0$ for points outside $T$, then I can use one of the Cauchy Theorems and conclude that $\int g_n=0$ and since $g_n \to g$  uniformly implies $g$ is continuous as well as $\int g_n \to \int g$, we can use Morera's Theorem and conclude that since $\int g=0$, $g$ is analytic. But what if $T$ is a triangular path around a hole in $G$? If we look at the winding number of such a point, then it is $1$ (say T goes around once counterclockwise). This is kind of where I get stuck in the proof because the assumptions of Morera's theorem don't hold. Anyway of clearing this up?
 A: Being analytic is a local property. So we only consider open balls contained in $G$ in the proof of Morera's Theorem. Then such situations as I described don't cause a threat to the proof mentioned. 
A: Careful! Just because a path winds around a hole, this doesn't mean the integral of your function $f$ around this point is nonzero. For example, consider $z^{-2}$ in $B(0,1)^{\times}$. Then the integral of $z^{-2}$ around a small circle around the origin is $0$, precisely beacuse $z^{-2}$ has $-z^{-1}$ as a primitive. In fact, we know that for a region $G$

A continuous function $f:G\to\Bbb C$ admits a primitive if and only if for every closed path $\gamma$ in $G$ we have $\int_\gamma f=0$.

Say that $f$ is locally integrable if every point in $G$ admits a nbhd where $f$ has a primitive. This is equivalent to the fact we can cover $G$ with open sets such that $f$ integrates to $0$ around any closed path inside such set. Morera's theorem says

A continuous function $f:G\to \Bbb C$ that is locally integrable is holomorphic.

Then of course there is no trouble, as you noted: suppose $G$ is  a region and $f_n$ is a sequence of holomorphic functions that converge uniformly to $f$ over any compact $K$ in $G$. We can cover $G$ with open balls around each of its points. Now take a closed triangle inside this ball. We can shrink the ball slightly to make it closed, and such that the triangle still fits there (why?) and since in this closed ball convergence is uniform, we have that $$\int_{\partial\Delta} f=\lim_{n\to\infty}\int_{\partial \Delta}f_n=0$$
This means $f$ is locally integrable, so it is holomorphic by Morera.
