12.3 Fourier Cosine and Sine SeriesOdd function

)()( xfxf

0)( p

pdxxf

xsin

Even function)()( xfxf

pp

pdxxfdxxf

0)(2)(

xcos(a) The product of two even functions is even.(b) The product of two odd functions is even.(c) The product of an even function and an odd function is odd.(d) The sum ( difference) of two even functions is even.(e) The sum ( difference) of two odd functions is odd.

Properties of Even/Odd Functions

If f(x) is an odd function on the interval (-p, p )

p

p

odd

pn

n dxxxfp

a cos)(1

0)(10

p

pdxxf

pa

0

Case1: Odd If f(x) is an even function on the interval (-p, p )

p

p

even

pn

n dxxxfp

a cos)(1

p

pdxxf

pa )(1

0

Case2: Even

p

p

even

pn

n dxxxfp

b

sin)(1p

even

pn dxxxf

p 0sin)(2

p

p

odd

pn

n dxxxfp

b

sin)(1 0

p

dxxfp 0

)(2

p

even

pn dxxxf

p 0cos)(2

12.3 Fourier Cosine and Sine SeriesFourier cosine Series

p

p dxxfa0

20 )(

p

pn

pn xdxxfb0

2 sin)( p

pn

n xdxxfp

a0

cos)(2

The Fourier series of an even function on the interval ( -p, p) is the cosine series

Fourier sine Series

The Fourier series of an odd function on the interval ( -p, p) is the sine series

Expand f(x) = x, -2 < x< 2, in a Fourier series.

Example

nb

n

n

1)1(4

the series converges to the function on ( -2, 2) and the periodic extension (of period 4)

Example: Expand in a Fourier series

nb

n

n)1(12

xx

xf01

01)(

12.3 Fourier Cosine and Sine Series

p

pn

pn xdxxfb0

2 sin)(

Fourier sine Series

The Fourier series of an odd function on the interval ( -p, p) is the sine series

is odd on the interval.

Gibbs phenomenon

-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

xx

xf01

00)(

)sin())1(1(21)(

1

nxn

xFSn

n

Example:15

term

s25

term

s

-3 -2 -1 1 2 3

0.2

0.4

0.6

0.8

1

125

term

s

Remark:1) The graphs has spikes near the discontinuities at ,,0 x

2) This behavior near a point of discontinuity does not smooth out but remains, even when the value n is taken to be large.

3) This behavior is known as the Gibbs phenomenon

Half-Range ExpansionsHalf-Range Expansions

All previous examples the function defined on ( -p, p)

we are interested in representing a function that is defined on an interval (0, L)

This can be done in many different ways

)(xf

1 2The cosine and sine series obtained in this manner are known as half-range expansions.

Half-Range Expansions

Half-Range ExpansionsHalf-Range Expansions

All previous examples the function defined on ( -p, p)

we are interested in representing a function that is defined on an interval (0, L)

This can be done in many different ways

)(xf

1 2 3

2L-periodic even extension 2L-periodic odd extension

L-periodic extension

Half-Range Expansions

Lxxxf 0 ,)( 2Example:

Expand (a) in a cosine series, (b) in a sine series, (c) in a Fourier series