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
