 Earlier, we gave a hand-waving argument that supported the Cauchy integral formula, let's prove it. Now, you might wonder why we're bothering to prove something that we believe is true, and the key to remember is the value of proof is the journey, not the destination. And in particular, what we have to do when we prove something is we have to review all of the things we should know at this point. Now, earlier we showed that if a simple closed curve C included places where f of z was not analytic and C prime is inside C, then the integral around C is the integral around C prime. We can contract our curve, in other words. Since we're assuming that f of z is analytic inside C, then f of z over z minus z naught will be analytic everywhere except at z naught. So we can contract our curve C down to a small circle C prime that surrounds the bad place. Let C prime be parameterized for r fixed but small and theta between 0 and 2 pi. So let C prime be parameterized, and so d z will be, and so our integral will be. Since r and z naught are constants, then f of z naught plus r e to the i theta is just a function of theta. So we can rewrite our function as, where g r theta h r theta correspond to the real and complex parts of our function and their functions of theta only. And since f of z is analytic, g r theta h r theta are necessarily continuous with continuous derivatives. So our integral becomes, now since we have a real integral and the integrand is continuous, the mean value theorem for integrals guarantees that we can evaluate this integral by evaluating our function at some point in the interval. So our original integral is this integral, which we can evaluate as a single complex number. So now let's take the limit as r goes to zero. Remember r is the radius of the circle around our bad point. While the specific values of theta prime and theta double prime will change, as r goes to zero, g r theta goes to the real part of f of z naught, while h r theta goes to the imaginary part of f of z naught. And so we get, which completes the proof of Cauchy's integral formula. Cauchy's integral formula is a remarkable result whose importance cannot be overstated because it says the value of an analytic function at any point inside a region is completely determined by the values along the border. As a simple example, suppose f of z is zero at every point along a closed curve c, and f of z is analytic in a connected region, including c, then f of z has to be zero everywhere inside our closed curve. By a Cauchy's integral formula, f of z naught is, by assumption, since along c, f of z is equal to zero, this integral simplifies to, now everybody who studies mathematics should know a couple of jokes, and one of them is, how many mathematicians does it take to screw on a light bulb? She gives it to a physicist, reducing it to the previous joke. So in the previous joke, I mean result, the function was zero at every point along our closed curve. Suppose it's equal to some constant at every point along a closed curve, then the same result applies, the function is equal to that constant everywhere inside our curve. And we can reduce it to the previous problem by defining a new function, g of z is f of z minus a, then g of z is zero everywhere along our curve, and by the previous result, everywhere inside c, we have g of z equal to zero, and so f of z is equal to a.