Importance of a region being convex So I have been reading and solving problems from complex analysis ( A.R Shastri's book), many times we talk of certain results in the case of convex regions. Is there something special about convex regions?
One example is:

Cauchy’s Theorem on a Convex Region: Let U be a convex
  region and A be a discrete subset of U. Let f be a continuous function
  on U and complex differentiable on U \ A. Then for any closed contour
  ω in U we have, $\int_{\omega}f(z)dz=0$

 A: I want to go a slightly different direction than the comments above and talk about why it's often useful to prove results on convex sets even if they hold on more general regions. The bottom line is that one can often reduce general cases to the convex case. So working on convex sets is a convenient simplification that often ends up implying general cases once enough machinery is developed. 
Many results in complex analysis which hold on convex sets $\Omega$, such as Cauchy's theorem, existence of globally defined harmonic conjugates on $\Omega$, that every holomorphic function can be uniformly approximated on compact sets by polynomials on $\Omega$, and existence of logarithms for all nonvanishing holomorphic functions on $\Omega$ actually hold if and only if $\Omega$ is simply-connected. This is a far more general condition than $\Omega$ being convex.
But there are ways one can go from the convex case to the simply-connected case. Of particular note is the Riemann mapping theorem, which states that if $\Omega$ is a simply connected and proper subset of $\mathbb{C}$, there is a biholomorphic map $f$ from $\Omega$ to the unit disk. So if a property is invariant under pre-composition with biholomorphic maps, one can prove the property on convex sets (or just the unit disk) and get the result for all simply connected, proper regions. Another common method one can use to go from the convex case to the simply connected case is the monodromy theorem, which allows one to conclude the existence of an analytic continuation on a simply connected region from analytic continuations along paths (which can often be made from results on open balls).
Another use proving things on convex sets is that one can then lift qualitative versions of some results to compact sets by taking finite covers with balls. One nice use of this technique is to prove versions of Harnack's inequality on general compact subsets of open regions $\Omega$.
