8 3 fourier series

PART 8.3: FOURIER SERIES X.1

FOR A DEFINED PERIOD FUNCTION:

on − L ≤ x < L with a period of 2L

a0 ∞ nπx nπx fW ( x) = + ∑ a n cos( ) + bn sin( ) 2 n=1 L L a πx mπx πx mπx 2πx 2πx fW ( x) = 0 + a1 cos( ) + a 2 cos( ) + ... + am cos( ) + b1 sin( ) + b2 sin( ) + ... + bm sin( ) 2 L L L L L L L 1 nπx a n = ∫ f ( x) cos( )dx n = 0,1,2,..., m L −L L

1 nπx bn = ∫ f ( x) sin( )dx n = 1,2,..., m L −L L L

Converges to:

X.2

f (x )   + f x + f x−   2

( ) ( )

at all points x where f is continuous at all points x where f is discontinuous

FOR A GENERAL FUNCTION ON A FINITE DOMAIN:

a0 ∞ + ∑ ak cos(kx) + bk sin(kx) 2 k =1 a fW ( x) = 0 + a1 cos( x ) + a 2 cos(2 x) + ... + an cos(nx) + b1 sin( x) + b2 sin(2 x ) + ... + bn sin(nx) 2 2π 1 ak = ∫ f ( x) cos(kx)dx k = 0,1,2,..., n fW ( x) =

π

bk = X.3

1

π

0 2π

∫ f ( x) sin(kx)dx

k = 1,2,..., n

0

EXTENTIONS:

Fourier Cosine Series: For a periodic function on − L ≤ x < L and f being an even function a0 ∞ nπx + ∑ an cos( ) 2 n=1 L a 2πx πx mπx ) + ... + a m cos( ) fW ( x) = 0 + a1 cos( ) + a2 cos( 2 L L L

f W ( x) =

2 nπx f ( x) cos( )dx n = 0,1,2,...m ∫ L0 L L

an =

bn = 0 n = 1,2,..., m

Fourier Sine Series: For a periodic function on

− L ≤ x < L and f being an odd function

∞

fW ( x) = ∑ bn sin( n =1

fW ( x) = b1 sin(

πx L

nπx ) L

) + b2 sin(

a n = 0 n = 0,1,2,..., m

2πx mπx ) + ... + bm sin( ) L L

2 nπx f ( x) sin( )dx n = 1,2,..., m ∫ L0 L L

bn =

Even Extension:

For a function defined on 0 ≤ period 2L

x < L or 0 < x ≤ L can be extended to an even period function with

a0 ∞ nπx + ∑ an cos( ) 2 n=1 L a πx 2πx mπx fW ( x) = 0 + a1 cos( ) + a2 cos( ) + ... + a m cos( ) 2 L L L

f W ( x) =

2 nπx an = ∫ f ( x) cos( )dx n = 0,1,2,...m L0 L L

bn = 0 n = 1,2,..., m

Odd Extension:

For a function defined on 0 ≤ period 2L ∞

fW ( x) = ∑ bn sin( n =1

fW ( x) = b1 sin(

πx L

nπx ) L

) + b2 sin(

a n = 0 n = 0,1,2,..., m

2πx mπx ) + ... + bm sin( ) L L

2 nπx f ( x) sin( )dx n = 1,2,..., m ∫ L0 L L

bn =

x < L or 0 < x ≤ L can be extended to an odd period function with