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Mathematics for NST Part IA Cambridge University, 2012-2013

Notes for Fourier Series Periodic functions: A periodic function is one which repeats regularly after a certain interval. The repetition may occur in space or time, or both. Such a function is termed 'oscillatory', and has the general form:

f ( x +P )=f (x) For all

x , where

P is a period of the function - an interval over which f ( x) can be

translated, and brought back into coincidence with itself. Least period: The least period of a periodic function is the smallest value of

P for which the above

relation holds. It is usually simply referred to as the period of the function. Trigonometric periodic functions and Fourier series: The functions

*

*

sin x and cos x

are periodic functions with least period

2p :

A Fourier Series is a series which can be used to approximate almost any periodic function as a sum of sines and cosines. Fourier space is a space for which the basis vectors are sines and cosines.

Functions as bases: Like vectors, functions can form a basis for an infinite-dimensional space of functions. Any function in the space can be represented as a linear combination of basis functions. As for vectors, orthogonal and orthonormal bases are particularly nice to work with; sets of functions which are mutually orthogonal are therefore useful as bases. On any interval

[a ,b ] , there may be many sets of functions that are orthogonal.

Orthogonality of trigonometric functions: The trigonometric functions

R { e inx } =1,cos x , cos 2x , cos 3x ... and

I { einx } =sin x , sin 2x , sin 3x ... are mutually orthogonal over the interval [-p , p ] . Since they are also invariant under translation by integer multiples of their least period

2p , orthogonality over

this interval implies orthogonality over all space:

p

0, m=n=0

p , m=n[?] 0 0,m [?] n

p

2p , m=n=0

p , m=n [?] 0 0, m [?]n

p

0, m=n=0 0, m=n[?] 0 0, m[?] n

[?] sin(mx)sin(n x) dx=

[?] cos(mx)cos (nx) dx=

[?] cos(mx) sin(nx) dx=

-p

-p

-p

Mathematics for NST Part IA Cambridge University, 2012-2013 Where orthogonality of two functions is defined by their inner product, as above. Negative values of

n need not be considered, since cosines and sines are symmetric (

cos( nx)=cos(-nx ) ) and antisymmetric ( sin(nx )=-sin(-nx ) ), respectively. Trigonometric functions therefore form an orthogonal basis which can be normalised to form an orthonormal basis by dividing the sines by

[?]p

and the cosines by

[?] 2p

.

Mathematics for NST Part IA Cambridge University, 2012-2013 Proof of mutual orthogonality: Demonstrate individual orthogonality:

p

[?] sin(nx)dx=

-p

p

[

[

[?] cos(nx )dx=

-p

]

p

-1 cos (nx ) =0 n

-p

1 sin ( nx ) n

]

p

-p

p

p

=0, [?] cos(0x )dx= [ 1 ]- p =2p

-p

Extend to mutual orthogonality:

mx (),n ,m[?] Z : Combination of two sines: sin(nx ) and sin

p

p

-p

-p

[?] sin ( nx ) sin(mx)dx=[?] 12 ( cos ( m-n ) -cos ( m+n ) ) dx Since

p

m-n and m+ n are both integers:

0,m [?] n

p , m=n[?] 0 0, m=n=0

[?] sin ( nx ) sin(mx)dx=

-p

Combination of two cosines:

p

p

-p

-p

mx and (),n , m [?] Z : cos

cos( nx)

[?] cos ( nx ) cos (mx)dx=[?] 12 ( cos ( m-n ) +cos ( m+n ) ) dx Since

m-n

and

m+ n are both integers:

0, m[?] n

p , m=n [?] 0 2p , m=n=0

p

[?] cos ( nx ) cos (mx)dx=

-p

Combination of a sine and a cosine:

cos( nx)

mx and (),n , m [?] Z : sin

Mathematics for NST Part IA Cambridge University, 2012-2013

p

p

-p

-p

[?] cos ( nx ) sin (mx)dx=[?] 12 ( sin ( m-n ) +sin ( m+n ) ) dx Since

p

m-n

and

m+ n are both integers:

0, m [?] n 0, m=n [?] 0 0, m=n=0

[?] cos ( nx ) sin (mx)dx=

-p

Generalising orthogonality to the interval For orthogonality over the interval

[-L , L] :

[-L , L] , trigonometric functions must have least

sin period 2L , thus be of the form

( mpxL )

cos and

( npxL )

for

n , m [?] Z . In this

case, the orthogonality relations are: L

npx cos ( dx=0

[?] sin ( mpx L ) L )

-L

0, m [?]n

p , m=n [?] 0 0, m=n=0 L npx sin dx=

[?] sin mpx L L

-L

( ) ( )

And the relations hold over any interval

0, m [?] n

p ,m=n [?]0 2p ,m=n=0 L npx cos dx=

[?] cos mpx L L

-L

( ) ( )

[a ,b ] such that b-a=2nL, n [?] Z .

Mathematics for NST Part IA Cambridge University, 2012-2013

Fourier series: Fourier analysis is based on the result that trigonometric functions form a basis for a space of periodic functions with least period

2L . Any periodic function of period 2L can be

uniquely written as a linear combination of trigonometric functions:

f ( x )=

Where

[ ( )

( )]

a0 [?]

npx npx

+ [?] a n cos

+bn sin 2 n=1 L L

an

bn

and

1 2

are known as Fourier coefficients and the factor of

on the first

term is included for later convenience. Fourier coefficients: The Fourier coefficients represent the weighting of each basis function in the overall function. They are analogous to the components of a vector and can be determined in the same way - by taking the inner product of the overall function with each of its basis functions, just as vector components are found by taking the projection of the overall vector onto each of its basis vectors:

an

is found by taking the inner product of

the series with basis function L

L

cos

( npxL )

:

bn

is found by taking the inner product

sin

with the basis function L

( npxL )

L

:

npx dx=[?] an cos 2 dx

[?] f ( x ) cos npx L L

-L

-L

npx dx=[?] an sin 2 dx

[?] f ( x ) sin npx L L

-L

-L

an L 2npx

[?] 1+cos dx 2 -L L

an L 2npx

[?] 1-cos dx 2 -L L

( )

( )

( )

[

a L 2npx

n x+

sin 2 2np

L

(

L

[?] a n=

( ) (

)]

L

-L

=a n L

( )

[?] b n=

an

(

above to hold for

)]

L

=bn L

-L

1 dx

[?] f ( x ) sin npx L -L L

a0

the constant term in the Fourier series expansion must be expressed in this way:

)

a L 2npx

n x-

sin 2 2np

L L

1 dx

[?] f ( x ) cos npx L -L L

In order for the expression for

[

( )

( )

also, hence hold for all

a0 /2

n[?]0 ,

, explaining why it is

Mathematics for NST Part IA Cambridge University, 2012-2013 L

L

L

[?]

1 0px 1 1 npx npx a0 = [?] f (x )cos dx= [?] f ( x )dx = [?] c+ [?] an cos

+b n sin dx L -L L L -L L -L n=1 L L

( )

( )

L

L 1 1

[?] a 0= [?] c dx= [ cx ]-L=2c L -L L

[?] c=

Where

a0 2 c

is a general term for the constant in the Fourier series.

( )

Mathematics for NST Part IA Cambridge University, 2012-2013 Sine and cosine series: The Fourier series of symmetric ( f ( x )=f (-x) ) and antisymmetric ( f ( x )=-f (-x ) ) functions behave differently to those of other periodic functions, due to the symmetry properties of sines (antisymmetric) and cosines (symmetric). For functions with such

y -axis have the following properties:

symmetry, integrals about the Symmetric: L

Antisymmetric: 0

L

L

0 L

[?] f ( x ) dx=[?] f ( x ) dx +[?] f ( x ) dx

[?] f ( x ) dx=[?] f ( x ) dx +[?] f ( x ) dx

-L

-L

0 -L L

-L

-[?] f ( x ) dx +[?] f ( x ) dx

0 0

u=-x [?] du=-dx , f ( u ) =f (x ) :

Set L

L

L

0 0

L

L

[?] [?] f ( x ) dx=[?] -f ( u ) du+[?] f ( x ) dx 0

0 u is an arbitrary variable:

L

L

[?] [?] f ( x ) dx=2[?] f ( x ) dx

[?] [?] f ( x ) dx=0

0 -L

u=-x [?] du=-dx , f ( u ) =-f (x ) :

Set

-L

u is an arbitrary variable: L

0 L

[?] [?] f ( x ) dx=[?] f (u ) du+[?] f ( x ) dx

-L

L

-L

-[?] f ( x ) dx +[?] f ( x ) dx 0

0 -L

-L

Thus, the Fourier series of symmetric and antisymmetric functions are as follows: Symmetric:

f ( x ) sin

( npxL )

is

Antisymmetric:

antisymmetric: L

[?] b n=

f ( x ) cos

( npxL )

is

antisymmetric: L

1 npx f ( x ) sin dx=0

[?]

L -L L

( )

[?] a n=

Thus the series is a series of cosines only.

1 npx f ( x ) cos dx=0

[?]

L -L L

( )

Thus the series is a series of sines only.

The coefficients of sine and cosine series can be found by integrating over just half a period: Cosine series: symmetric:

f ( x ) cos

( npxL )

is

Antisymmetric: symmetric:

f ( x ) sin

( npxL )

is

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