Reddit reviews Complex Analysis: Stewart/Tall

I'm a fan of Ian Stewart and David Tall's book on the subject if you can get your hands on it.

One of the neat things is how you prove that the integral of a differentiable function around a closed loop is zero, if the function is defined everywhere inside that loop. You break up the area inside the loop into triangles, so that your integral is the sum of the integrals around each of those triangles. A differentiable function is one that's roughly linear at small scales, and linear functions have antiderivatives, and the integral of something with an antiderivative around a closed loop is 0 by the Fundamental Theorem of Calculus, so the integral around each small triangle is gonna be roughly 0. And so, adding all the triangles together, the total integral is 0.

(You need to keep careful track of the epsilons and such to make that rigorous, but the point is that the integral over each triangle is 1) small because the triangle is small and 2) small because it's roughly linear, so it's like doubly small. So it stays small when you add them all up)

I'm sorry, that wasn't totally coherent… but read the book, it'll make sense