2Fourier series

IB Methods

2.1 Fourier series

So. As mentioned in the previous chapter, we want to find a set of “basis

functions” for periodic functions. We could go with the simplest case of periodic

functions we know of — the exponential function

e

inθ

. These functions have a

period of 2

π

, and are rather easy to work with. We all know how to integrate

and differentiate the exponential function.

More importantly, this set of basis functions is orthogonal. We have

(e

imθ

, e

inθ

) =

Z

π

−π

e

−imθ

e

inθ

dθ =

Z

π

−π

e

i(n−m)θ

dθ =

(

2π n = m

0 n 6= m

= 2πδ

nm

We can normalize these to get a set of orthonormal functions

n

1

√

2π

e

inθ

: n ∈ Z

o

.

Fourier’s idea was to use this as a basis for any periodic function. Fourier

claimed that any f : S

1

→ C can be expanded in this basis:

f(θ) =

X

n∈Z

ˆ

f

n

e

inθ

,

where

ˆ

f

n

=

1

2π

(e

inθ

, f) =

1

2π

Z

π

−π

e

−inθ

f(θ) dθ.

These really should be defined as

f

(

θ

) =

P

ˆ

f

n

e

inθ

√

2π

with

ˆ

f

n

=

e

inθ

√

2π

, f

, but for

convenience reasons, we move all the constant factors to the

ˆ

f

n

coefficients.

We can consider the special case where

f

:

S

1

→ R

is a real function. We

might want to make our expansion look a bit more “real”. We get

(

ˆ

f

n

)

∗

=

1

2π

Z

π

−π

e

−inθ

f(θ) dθ

∗

=

1

2π

Z

π

−π

e

inθ

f(θ) dθ =

ˆ

f

−n

.

So we can replace our Fourier series by

f(θ) =

ˆ

f

0

+

∞

X

n=1

ˆ

f

n

e

inθ

+

ˆ

f

−n

e

−inθ

=

ˆ

f

0

+

∞

X

n=1

ˆ

f

n

e

inθ

+

ˆ

f

∗

n

e

−inθ

.

Setting

ˆ

f

n

=

a

n

+ ib

n

2

, we can write this as

f(θ) =

ˆ

f

0

+

∞

X

n=1

(a

n

cos nθ + b

n

sin nθ)

=

a

0

2

+

∞

X

n=1

(a

n

cos nθ + b

n

sin nθ).

Here the coefficients are

a

n

=

1

π

Z

π

−π

cos nθf(θ) dθ, b

n

=

1

π

Z

π

−π

sin nθf(θ) dθ.

This is an alternative formulation of the Fourier series in terms of sin and cos.

So when given a real function, which expansion should we use? It depends. If

our function is odd (or even), it would be useful to pick the sine/cosine expansion,

since the cosine (or sine) terms will simply disappear. On the other hand, if we

want to stick our function into a differential equation, exponential functions are

usually more helpful.