Chapter 6

Fourier Series

Key Results

•

Derive formulas for Fourier coefficients•

Calculate the Fourier series of a periodic function

•

Use Fourier series to approximate wave functions•

Approximate mathematical constants

using series

•

Half range expansions

Periodic Functions

A function f (x) is called periodic

if•

f

is defined for all real values x

•

there is a positive number p

such thatf (x

+ p) = f (x) for all x.

The number p

is called a period

of f.

Graph of a Periodic Function

The graph of a periodic function can be obtained by periodic repetition of a portion

of its graph over an

interval of length p.

x x + p

f

(x

+ p) = f

(x)

Examples

For a positive constant k, the functionsf (x) = sin kx

and g(x) = cos

kx

But polynomials (e.g. x, x2, x3), exponential functions (e.g. ax), logarithmic functions (e.g. loga

x) are not periodic functions.

Some Properties

If f

has period p, then

Inductively, for any positive integer n,

Thus, 2p, 3p, …

are also periods of f.

X X+ p

If f

and g

have period p, and a

and b

are constants, then the function

h(x) = af

(x) + bg(x)satisfies

h(x

+

p) = af

(x + p) + bg(x + p)= af

(x) + bg(x)

= h(x)for all x. That is, h

is also periodic with period p.

Even FunctionsA function f

is an even function

if f (x) = f (x).

e.g. f (x) = x2.

Check:

The graph of an even function is symmetrical about the y-axis.

Integration property:

y

= x2

f

(−

x) = f

(x)

symmetrical about

y-axis

= 2

Even Functions

(cont’d)Further examples of even functions:

e.g.

e.g.

f

(x) = cos

kx

where k

is a constant

f

(x) = x4

−

x2

Odd FunctionsA function f

is an odd function

if f (x) =

f (x).

e.g. f (x) = x3.

Check:

The graph of an odd function is symmetrical about the origin.

Integration property:

f

(−

x) = −

f

(x)

y

= x3symmetrical about origin

areas of regions areof same magnitude but of different signs

= 0

Odd Functions

(cont’d)Further examples of odd functions:

e.g.

e.g.

f

(x) = sin kx

where k

is a constant

f

(x) = x5

−

x

Product Propertiesf (x)

odd

f (x) g(x)g(x)

odd odd

odd

eveneven

even

even

even

e.g.

is even

odd even odd

Trigonometric SeriesAim to represent periodic functions using simple

periodic functions1, cos

x, sin x, cos

2x, sin 2x, …, cos

nx, sin nx, …

combined into a series

called a trigonometric series.

Fourier SeriesSince each of the terms

1, cos

x, sin x, cos

2x, sin 2x, …, cos

nx, sin nx, …is of period 2 , it follows that if the series

converges, its sum will be a periodic function f

of period 2 .

Write

Conversely, given a periodic function f

of period 2 , find coefficients a0, a1, a2, a3, …, b1, b2, b3, … such that

The series on the right side of the equation is called the Fourier series

of f.

Jean Baptiste

Joseph Fourier(1768 –

1830)

Determine a0

Recall

Integrate both sides:

Consider term-by-term integration

on the right side.

Determine am

, m

> 0

Recall

Multiply

both sides by cos

mx

and integrate:

Examine the three integrals on right side, one at a time.

First integral

0 0

Third integral easier than second integral, consider third integral now

sin nx

is an odd function and cos

mx

is an even function.

Thus, the product sin nx

cos

mx

is an odd function.

Second integral

cos

2mx cos

0 = 1

sin kπ

= 0k

= 0, ±1, ±2, ±3, …

now simplifies

= am

Determine bm

, m

> 0

Recall

Multiply

both sides by sin mx

and integrate:

Examine the three integrals on right side, one at a time.

First integral

sin mx

is an odd function

Second integral

cos

nx

is an even function and sin mx

is an odd function.

Thus, the product cos

nx

sin mx

is an odd function.

Third integral

sin kπ

= 0k

= 0, ±1, ±2, ±3, …

cos

2mxcos

0 = 1

Euler FormulasA periodic function f(x) of period 2 with Fourier series

has Fourier coefficients given by

These formulas are known as Euler formulas.

Leonhard Euler(1707 –

1783 )

ExampleConsider

f

(x) is piecewise continuousOmitting a single point (x

value) does not affect integrals

Leave f

undefined at x

= 0, x

= ±π

Over the interval (– , ), graph is symmetrical about the origin, f is an odd function.

Now, f(x) cos

nx

is also an odd function. Thus,

f(x),

sin nx

both odd functionsf(x)

sin nx

is an

even function

(page 10)only these terms left

n

= 1

n

= 5

n

= 4n

= 3

n

= 2

An Approximation for

From

let

At Points of DiscontinuityAt points of discontinuity of f, the Fourier series has

sum equal to the average of the left limit and right limit.

For example, at x

= 0,

Check indeed that the average of –

k

and k

is 0.

page 11

General Period p

= 2LLet f(x) be a periodic function of period p

= 2L.

To find the Fourier series of f(x), use a substitution (change of variable) in the Euler formulas on page 10.

General Fourier Formulas

(derivation)Fourier series of g(v) (from page 10, v

for x, g

for f

):

period 2

General Fourier Formulas

(derivation)Fourier series of g(v) (from page 10, v

for x, g

for f

):

period 2

General Fourier Formulas

Period p

= 2L

General Fourier Formulas

Period p

= 2L Let L

=

These are the Euler formulas on page 10

Example

Considerp

= 2L

= 4

i.e. L

= 2

Graph is symmetrical about y-axis, i.e. f(x) is an even function

L

= 2

Fourier Cosine and Sine SeriesThe Fourier series of an odd function

or an even

function

simplifies to only involve sine terms

or cosine terms.

Simplification comes from the following key integration properties

Fourier Cosine SeriesThe Fourier series of an even function f(x)

of period 2L

is

becausef(x) is an even function

Fourier cosine series

Fourier Cosine Series

(cont’d)Also,

f(x) is an even function

Fourier Sine SeriesSimilar to the development of the Fourier cosine series,

there is a Fourier sine series

for an

odd function f(x) of period 2L.

Exercise: Using ideas involving integrals of even functions and odd functions, show that an

= 0 for n

= 0, 1, 2, …

Sum and Scalar Multiplication

The Fourier coefficients of f1

+ f2

are the sums of the corresponding Fourier coefficients

of f1

and f2

.

For any constant c, the Fourier coefficients of cf

are c times the corresponding Fourier coefficients

of f.

Saw Tooth Function

π

=

The cosine half range expansion is

where

ExampleFind the cosine half range expansion of

L

= π

0

0

0

The sine half range expansion is

where

End of Chapter 6