12.2 Fourier Series

Trigonometric Series

,sin,sin,sin

,cos,cos,cos,1

32

32

xxx

xxx

ppp

ppp

is orthogonal on the interval [ -p, p].

In applications, we are interested to expand a function f(x) defined on [-p, p] as a linear combination

xbxaa

xFSp

nn

np

nn

sincos2

)(1

0

p

p p

nn xdxxf

pa cos)(

1

p

p p

nn xdxxf

pb sin)(

1

p

pdxxf

pa )(

10

Fourier series of the function f

Fourier coefficients of f

12.2 Fourier Series

xx

xxf

0

00)(

Example:

xbxaa

xfp

nn

np

nn

sincos2

)(1

0

p

p p

nn xdxxf

pa cos)(

1

p

p p

nn xdxxf

pb sin)(

1

p

pdxxf

pa )(

10

Fourier seriesExpand in a Fourier series

2

)1(1

na

n

n

nbn

120

a

nxnxxFSn

nn

n

sincos4

)( 1

1

)1(12

12.2 Fourier Series

Example:

xbxaa

xfp

nn

np

nn

sincos2

)(1

0

p

p p

nn xdxxf

pa cos)(

1

p

p p

nn xdxxf

pb sin)(

1

p

pdxxf

pa )(

10

Fourier seriesExpand in a Fourier series

nb

n

n

)1(1 0na

10 a

x

xxf

01

00)(

)sin())1(1(

2

1)(

1

nxn

xFSn

n

Convergence of a Fourier Series

f(x) is piecewise continuous on the interval [-p,p]; if f(x) is continuous except at a finite number of points in the interval and have only finite discontinuities at these points.

piecewise continuous

pp

Theorem 12.2.1 Conditions for Convergence

' , ff piecewise continuous on [-p,p]

is a point of continuity.

is a point of discontinuity.

denote the limit of f at x from the right and from the left

12.2 Fourier Series

xx

xxf

0

0)(

Example: Expand in a Fourier series

nxnxxFSn

nn

n

sincos4

)( 1

1

)1(12

)()( xfxFS ),0()0,( x

2

)0()0()0( ffFS 0x

2

Remark:

Sequence of Partial Sums

-3 -2 -1 1 2 3

0.2

0.4

0.6

0.8

1

3

)3sin(2)sin(2

2

1 xx

-3 -2 -1 1 2 3

0.2

0.4

0.6

0.8

1

7

)7sin(2

5

)5sin(2

3

)3sin(2)sin(2

2

1 xxxx

-3 -2 -1 1 2 3

0.2

0.4

0.6

0.8

1

-3 -2 -1 1 2 3

0.2

0.4

0.6

0.8

1

x

xxf

01

00)(

)sin())1(1(

2

1)(

1

nxn

xFSn

n

Example:15

ter

ms

25 t

erm

s

Sequence of Partial Sums

-3 -2 -1 1 2 3

0.2

0.4

0.6

0.8

1

-3 -2 -1 1 2 3

0.2

0.4

0.6

0.8

1

x

xxf

01

00)(

)sin())1(1(

2

1)(

1

nxn

xFSn

n

Example:

15 t

erm

s

25 t

erm

s

-3 -2 -1 1 2 3

0.2

0.4

0.6

0.8

1

-3 -2 -1 1 2 3

0.2

0.4

0.6

0.8

1

125

term

s

1000

ter

ms

MATHEMATICA Plot[0.5+Sum[ (1-(-1)^n)*Sin[n x]/(n Pi),{n,1,1000}],{x,-Pi,Pi}];

Periodic Extension

xx

xxf

0

00)(

Example: Consider the funciion

Periodic extension of the function f

3 55 3

12.2 Fourier Series

Example: Consider the function

x

xxf

01

00)(

3 435

Periodic extension of the function f

12.2 Fourier Series

Example: Consider the function

x

xxf

01

00)(

3 435

Periodic extension of the function

)sin())1(1(

2

1)(

1

nxn

xFSn

n

a Fourier series not only represents the function on the interval ( -p, p) but also gives the periodic extension of f outside this interval.

3 435

)(xFS

2p is the fundamental period

)(xf

Periodic Extension

xx

xxf

0

00)(

Example: Consider the funciion

3 55 3

nxnxxFSn

nn

n

sincos4

)( 1

1

)1(12

3 55 3

3 55 3

(A)

(B)

(C)

Which one represents FS(x) ?