We start this section off with a surprisingly important and useful theorem.  In real analysis we learned that power series were extremely important for providing insight into functions of real variables in theorems such as the Stone-Weierstrass Theorem.  In ordinary differential equations, we saw the versatility of the application of power series to solving differential equations that could not easily be solved otherwise.  Here, we discover the natural object to study in the complex plane is not the power series, but rather the Laurent series and we then find immediate application in the Cauchy Residue Theorem.