Chapter 2 Functions from C to C : Some Easy Examples

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The complex numbers form what Mathematicians call (for no very good

reason) a _eld, which is a collection of things you can add, subtract, multiply

and (except in the case of 0) divide. There are some rules saying precisely

what this means, for instance the associativity 'laws', but they are just the

rules you already know for the real numbers. So every operation you can do

on real numbers makes sense for complex numbers too.

After you learnt about the real numbers at school, you went on to discuss

functions such as y = mx + c and y = x2. You may have started o_ by

discussing functions as input-output machines, like slot machines that give

you a bottle of coke in exchange for some coins, but you pretty quickly went

on to discuss functions by looking at their graphs. This is the main way of

thinking about functions, and for many people it is the only way they ever

meet.

Which is a pity, because with complex functions it doesn't much help.

The graph of a function from R to R is a subset of R_R or R2 . The graph of

a function from C to C will be a two-dimensional subset of C _C which is a

surface sitting in four dimensions. Your chances with four dimensional spaces

are not good. It is true that we can visualise the real part and imaginary

part separately, because each of these is a function from R2 to R and has

graph a surface. But this loses the relationship between the two components.

So we need to go back to the input-output idea if we are to visualise complex

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34 CHAPTER 2. EXAMPLES OF COMPLEX FUNCTIONS

Figure 2.1: The random points in a square

functions.

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