Example. Solution - UCL
[Pages:6]Continuing from the last lecture.... Example. (ii) Let f be a periodic function of period 2 such that
f (x) = 2 - x2 for - x < Solution: So f is periodic with period 2 and its graph is:
We first check if f is even or odd. f (-x) = 2 - (-x)2 = 2 - x2 = f (x), so f (x) is even.
Since f is even,
bn = 0 2
an = 0 f (x) cos(nx)dx
Using the formulas for the Fourier coefficients we have
2
an = 0 f (x) cos(nx)dx
2 =
(2 - x2) cos(nx)dx
0
=
2 (
(2 - x2) sin nx n
0
-
sin nx
-2x
dx)
0
n
=
2 ([(2
- 2) sin n
- (2
sin 0 - 0) ] +
sin nx
2x
dx)
n
n
0
n
22
=
x sin(nx)dx
n 0
22 =(
cos nx -x
-
cos nx
(-
)dx)
n
n0
0
n
=
2
2 n
1 ([-
n
cos n]
+
[
sin nx n2
]0
)
4 = - cos n
n2
=
-4/n2 if n is even 4/n2 if n is odd
It remains to calculate a0.
2 a0 =
(2 - x2)dx
0
=
2 [2x
-
x3 3
]0
42 =
3
The Fourier Series of f is therefore
1 f (x) = 2 a0 + a1 cos x + a2 cos 2x + a3 cos 3x + ...
b1 sin x + b2 sin 2x + b3 sin 3x + ...
22
1
1
1
1
= + 4(cos x - cos 2x + cos 3x - cos 4x + cos 5x + ....)
3
4
9
16
25
10.6 Functions with period 2L
If a function has period other than 2, we can find its Fourier series by making a change of variable. Suppose f (x) has period 2L, that is f (x + 2L) = f (x) for all x. If we let
x
Lt
t = and g(t) = f (x) = f ( )
L
then, g has period 2 and x = ?L corresponds to t = ?. We know the Fourier series of g(t):
1
2 a0 + (an cos nt + bn sin nt).
n=1
where
1
a0
=
f (t)dt
-
1
an
=
f (t) cos(nt)dt
-
1
bn
=
f (t) sin(nt)dt
-
If
we
now
use
the
Substitution
Rule
with
x
=
Lt
,
then
t
=
x L
,
dt
=
(
L
)dx,
we have the following:
Definition. If f (x) is a piecewise continuous function on [-L, L], its Fourier
Series is
1
nx
nx
2 a0 + an cos( L ) + bn sin( L ),
n=1
where
1L
a0
=
L
f (x)dx
-L
1L
nx
an
=
L
f (x) cos( )dx
-L
L
1L
nx
bn
=
L
f (x) sin( )dx
-L
L
Example. Find the Fourier series of the triangular wave function defined by f (x) = |x| - a for -a x a and f (x + 2) = f (x) for all x.
Solution: So f is periodic with period 2a and its graph is:
We first check if f is even or odd.
f (-x) = | - x| - a = |x| - a = f (x), so f (x) is even.
Therefore,
bn = 0
2a
nx
an = a 0 f (x) cos( a )dx
Using the formulas for the Fourier coefficients we have
2a
nx
an = a
(|x| - a) cos(
0
a
)dx
2a
nx
= (x - a) cos( )dx
a0
a
=
2 (
a
(x
-
a)
sin(
nx a
n
)
a
a-
0
a
a
sin(
nx a
n
)
dx)
0
a
2 a2 =(
nx cos( )
a)
a n22
a0
2a = (cos n - 1)
n22
0 if n is even
=
4a
-
if n is odd
n22
It remains to determine a0.
2a a0 = a 0 |x| - adx
2a
=
x - adx
a0
=
2 x2 [
a2
-
ax]a0
= -a
The Fourier Series of f is therefore
f (x) = a0 + a1 cos x + a2 cos 2x + a3 cos 3x + ...
a 4a x 1 3x 1 5x
= - - (cos( ) + cos( ) + cos( ) + ...)
2 2
a9
a 25
a
10.7 Parseval's Result
The folowing result can be used to find the sum of certain series from calculated Fourier series.
Theorem Let f be a bounded function defined in the interval [-L, L] with at most a finite number of maxima and minima and at most a finite number of discontinuities in the interval [-L, L], and associated to the Fourier series
1
2 a0 + (an cos nx + bn sin nx).
n=1
Then
1 L
L
(f (x))2dx
-L
=
1 2
a20
+
(a2n
n=1
+
b2n).
Example. Let f be a periodic function defined by f (x) = x for all - x . Use Parseval's result to show that
2 1
=
.
6
n2
n=1
Solution: We have shown that
2
2
2
2
f (x) = 2 sin x - sin 2x + sin 3x - sin 4x + sin 5x + ...
2
3
4
5
Using Parseval's Identity
1
(f (x))2dx
-
=
1 2
a20
+
(a2n
n=1
+ b2n)
1
x3 3
-
=
4(1
+
1 4
+
1 9
+
...)
22
11
= 4(1 + + + ...)
3
49
2
11
= 1 + + + ...
6
49
2 1
= 6
n2
n=1
................
................
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