 For real variables, we have the intermediate value theorem, if f of x is continuous over an interval, and m is somewhere between the values of the function at the endpoints, then there's some c in the interval where our function takes on the value m. This is also known as the Goldilocks principle. Somewhere between too small and too big is just right. We can do the mean value theorem in another way. If f of x is continuous over an interval, it's impossible for it to avoid any value between the function values at the endpoints. Unfortunately, between is meaningless for complex numbers, but we have a stronger property for complex functions, so we might ask what if f of z is analytic in some connected region R? Now, we'll introduce a useful simplification for the following, and in fact, for much of what we're going to do after this point, we'll only need f of z to be analytic on and inside some circle. This allows us to introduce one generalization. The Cauchy integral formula requires f of z to be analytic in a simply connected region R. But now, consider z not in R. By assumption, our function is analytic at z equal to z not. So even if R is not simply connected, we can find a small circle around any point in R where f of z is analytic on and inside the circle. So let's see where this takes us. Suppose f of z is analytic in R, and z not is some point in R. f of z will be analytic on and inside some sufficiently small circle centered at z not. So using the Cauchy integral formula we'll have using z equal to z not plus R e to the i theta gives us, so here's another version where the value of an analytic function is going to be completely determined by its values along some closed curve. Well, let's look a little closer. Since the integral is over the interval between 0 and 2 pi, we can interpret the right side as the mean value of our function along the circle c, and this gives us the following. And we might call this the mean value property, though there's no standardized name for this result. If f of z is analytic in some region R, then f of z not is where f of z is analytic on and inside the circle. So let's take stock. The Cauchy integral formula tells us the value of an analytic function at z not is determined by its values along any closed curve surrounding z not. The complex mean value theorem tells us the value is limited by its values along a circle surrounding z not. And it's important to realize that these two tell us slightly different things. For example, suppose f of z is analytic on and inside a circle centered at z not. Moreover, suppose the value of f of z is real along our circle. What does the Cauchy integral formula tell you? And what does the complex mean value theorem tell you? So the Cauchy integral formula tells us the value of f of z not. Unfortunately, while f of z is real along c, the integrand will not be. So we can't really say much more about the function value. In contrast, the mean value theorem tells us now since the integrand is real, the integral will also be real. And that tells us the function value at z not will also be real.