 The Cauchy-Gorsuch theorem allows us to find many complex integrals quickly. However, it relies on the notion of a region inside a closed curve. Ancient philosophers refer to this as the donut problem. Is the donut hole inside the donut? In two dimensions, a donut, wait, an annulus is a region between two closed curves. Suppose r is an annulus, and f of z is analytic in r, but not in the hole. What do we know about our contour integral for a simple closed curve in r? So suppose our simple closed curve doesn't go around the hole. Then, by anyone's definition of inside, f of z is analytic inside of c, and so the integral is zero by the Cauchy-Gorsuch theorem. But what if c goes around the hole? To answer this question, let's consider two closed curves c1 and c2, where c1 goes around the hole but stays inside the annulus, and c2 defines the inner boundary. We'll assume they have the same orientation. In other words, they both go in the same direction, in this case counterclockwise. If we cut the annulus, the original path can't get back to its starting point without taking a detour c3 around the cut. So the new path can be described as c1 plus c3 minus c2 minus c3, where we subtract the portions where we're traveling in the opposite direction. And importantly, f of z is analytic inside this new curve by anyone's definition of inside, so Cauchy-Gorsuch applies. So by Cauchy-Gorsuch, the integral around this curve c1 plus c3 minus c2 minus c3 must be zero. By the additivity of the integral, we can break it up into each of its component pieces, and here our c3 and our minus c3 cancel each other out, and consequently, and this tells us the following, suppose c1 is any path around a hole in a region where f of z is analytic. If c2 marks the inner boundary of the hole, then the integral over c1 is equal to the integral over c2. But since c2 is the inner boundary, f of z is not analytic inside c2 by anyone's definition, so we don't know the value of the integral. Informally, we can say the curve c1 is contracted down to c2. For Cauchy-Gorsuch to apply, we have to be able to contract c1 down to a point. Otherwise, we'll get hung up on an inner boundary curve and be unable to say anything about the value of the integral, at least not yet. And we can introduce the following term. We say that a region is simply connected if every closed curve in the region can be contracted down to a point. And so we restate Cauchy-Gorsuch as the following. Suppose f of z is analytic in a simply connected region, r, then for any simple closed curve in r, the integral over that closed curve is zero.