2Fourier series

IB Methods

2 Fourier series

The first type of functions we will consider is periodic functions.

Definition

(Periodic function)

.

A function

f

is periodic if there is some fixed

R such that f(x + R) = f(x) for all x.

However, it is often much more convenient to think of this as a function

f

:

S

1

→ C

from unit circle to

C

, and parametrize our function by an angle

θ

ranging from 0 to 2π.

Now why do we care about periodic functions? Apart from the fact that

genuine periodic functions exist, we can also use them to model functions on a

compact domain. For example, if we have a function defined on [0

,

1], we can

pretend it is a periodic function on

R

by making infinitely many copies of the

function to the intervals [1, 2], [2, 3] etc.

x x