IB Complex Analysis - Introduction

0Introduction

IB Complex Analysis

0 Introduction

Complex analysis is the study of complex differentiable functions. While this

sounds like it should be a rather straightforward generalization of real analysis,

it turns out complex differentiable functions behave rather differently. Requir-

ing that a function is complex differentiable is a very strong condition, and

consequently these functions have very nice properties.

One of the most distinguishing results from complex analysis is Liouville’s

theorem, which says that every bounded complex differentiable function

C → C

must be constant. This is very false for real functions (e.g.

sin x

). This

gives a strikingly simple proof of the fundamental theorem of algebra — if the

polynomial

has no roots, then

is well-defined on all of

, and it is easy to

show this must be bounded. So p is constant.

Many things we hoped were true in real analysis are indeed true in complex

analysis. For example, if a complex function is once differentiable, then it is

infinitely differentiable. In particular, every complex differentiable function has

a Taylor series and is indeed equal to its Taylor series (in reality, to prove these,

we show that every complex differentiable function is equal to its Taylor series,

and then notice that power series are always infinitely differentiable).

Another result we will prove is that the uniform limit of complex differentiable

functions is again complex differentiable. Contrast this with the huge list of

weird conditions we needed for real analysis!

Not only is differentiation nice. It turns out integration is also easier in

complex analysis. In fact, we will exploit this fact to perform real integrals by

pretending they are complex integrals. However, this will not be our main focus

here — those belong to the IB Complex Methods course instead.