 So let's introduce what may be the most important theorem from complex analysis, the Cauchy integral formula. Recall we can express a function in terms of its Taylor series where the coefficients are given by the nth derivatives. Now suppose C is a simple closed curve in our region. Let's assume things and see where this takes us. So let's suppose our curve loops around Z0 where we're getting our Taylor expansion around. But let's consider f of Z divided by Z minus Z0. So if f of Z is our Taylor series then dividing by Z minus Z0 gives us a new series which we can write as or splitting off that first term and we can integrate. And again since we assumed things we'll assume that we can switch the order of integration and again assuming things we'll assume we can contract C down to a small circle which will allow us to parameterize Z and so our first integral becomes, but remember an is the nth derivative divided by n factorial. So a0 is going to be f of Z0 and so this first integral will be 2 pi f of Z0 i. Meanwhile if we consider the terms of this series they're all integrals of Z minus Z0 to some power and n is greater than or equal to 1 and it turns out we can compute these are all equal to 0 and that gives us a remarkable result if we find the value of this integral then multiply by 1 over 2 pi i we get f at the center. Now as it turns out the things we have to assume is only that f of Z is analytic inside our closed curve and this gives us the Cauchy integral formula. Suppose f of Z is analytic inside a simple closed curve C then the value at Z0 is determined by an integral. We'll actually prove the Cauchy integral formula later it turns out that the Taylor expansion is not the easiest way to do that because there's a whole bunch of other things we have to prove before we can use the Taylor expansion but for now note that the Cauchy integral formula allows us to evaluate contour integrals by finding the bad place Z0 inside our curve, rewriting our integrand, then evaluating a new function at a single point. So for example let's say we want to evaluate this integrand. So we note that the bad place for this integrand is at Z equal to 1 which is inside our curve. So we want f of Z so that our integrand is f of Z over Z minus 1. Comparing the two sides tells us to take f of Z equal to Z. We take f of Z equal to Z so f of Z0 will be this integral and if we want to take Z0 to be 1 we can compute what f of 1 is equal to and since we just want the integral we'll multiply by 2 pi i and putting things back where we found them f of Z was Z and so our integral is equal to 2 pi i.