Morera's Theorem in complex analysis i have been thinking about this problem and would appreciate it if anyone can enlighten me.  
Morera's Theorem states the following: Let $f(z)$ be a continuous function on a simply connected region, $D$. If for all loops in $D$, one has $\int_{C}f(z) dz=0$, then $f$ is holomorphic in $D$.
How can i also prove that the statement above is also true if $D$ is not simply connected? Thank you!
 A: $f$ is holomorphic on $D$ if and only if for each $x\in D$, $f$ is holomorphic on some open disk in $D$ containing $x$.  Open disks are simply connected, so the theorem you cited can be applied locally for any region.

I will elaborate.
Theorem (from your question): Let $f(z)$ be a continuous function on a simply connected region, $D$. If for all loops in $D$, one has $\int_{C}f(z) dz=0$, then $f$ is holomorphic in $D$.
Corollary: Let $f(z)$ be a continuous function on a region, $D$. If for all loops in $D$, one has $\int_{C}f(z) dz=0$, then $f$ is holomorphic in $D$.
Proof (that the corollary follows from the theorem):  Suppose $f$ and $D$ are as in the hypothesis of the corollary.  Let $z$ be a point in $D$.  Let $U$ be an open disk in $D$ containing $z$.  For all loops in $U$, by hypothesis $\int_{C}f(z) dz=0$.  Note that $U$ is simply connected, and $f$ is continuous on $U$, so by the theorem, $f$ is holomorphic on $U$.  Since $z$ was arbitrary and $f$ is holomorphic on a neighborhood of $U$, $f$ is holomorphic on $D$. 
