 Suppose g of z is analytic everywhere except at z equal to z0. If z is a simple closed curve whose interior includes z0, the Cauchy integral formula gives us a way to compute g of z. We can rewrite g of z as f of z over z minus z0 in effect factoring out the bad point. If f of z is analytic everywhere inside z, then the Cauchy integral formula applies, the integral can be evaluated by finding f of z0. Since f of z over z minus z0 is g of z, then the original integral is 2 pi i times f of z0. For example, let's find this integral. We want to find a function that's analytic inside z where our integrand is f of z over the bad point. Comparing the two sides suggests that f of z is e to the z squared. Unfortunately, this is analytic inside z, so the integral is easily computed. Wait, that's not actually unfortunate. We want that to happen. So let f of z equal e to the z squared. Then our integral becomes the Cauchy integral formula tells us that f of 1 is 1 over 2 pi times the integral. And so that says the integral is 2 pi i times f of 1. And so that tells us the value of the integral will be, or let's try to find this integral. The important thing to remember is that if it's not inside our closed curve, it doesn't matter. So first, let's plot c, the upper half circle of radius 5. Since sine z and z squared plus 1 are analytic everywhere, their quotient will be analytic everywhere that the denominator isn't 0. So solving for the denominator equal to 0 gives us sine z over z squared plus 1 is analytic everywhere except z plus or minus i. So again, remember, if it's not inside our closed path, it doesn't matter. While sine z over z squared plus 1 is analytic everywhere except z plus or minus i, only z equals i is inside our closed path. So we want to find f of z, where f of z over z minus i is sine z over z squared plus 1. Again, you can think about this as factoring out the bad point at z equal to i. We've factored z squared plus 1, and comparing the two sides suggests we use f of z equal to sine z over z plus i. So I think f of z equals sine z over z plus i gives us, and Cauchy's integral formula tells us this integral is 2 pi i times f at the bad point, which will be...