0Introduction

IB Complex Methods

0 Introduction

In Part IA, we learnt quite a lot about differentiating and integrating real

functions. Differentiation was fine, but integration was tedious. Integrals were

very difficult to evaluate.

In this course, we will study differentiating and integrating complex functions.

Here differentiation is nice, and integration is easy. We will show that complex

differentiable functions satisfy many things we hoped were true — a complex

differentiable function is automatically infinitely differentiable. Moreover, an

everywhere differentiable function must be constant if it is bounded.

On the integration side, we will show that integrals of complex functions can

be performed by computing things known as residues, which are much easier

to compute. We are not actually interested in performing complex integrals.

Instead, we will take some difficult real integrals, and pretend they are complex

ones.

This is a methods course. By this, we mean we will not focus too much on

proofs. We will at best just skim over the proofs. Instead, we focus on doing

things. We will not waste time proving things people have proved 300 years ago.

If you like proofs, you can go to the IB Complex Analysis course, or look them

up in relevant books.