Post on 20-Apr-2022
1
UNIVERSITY OF TECHNOLOGYElectromechanical Department
2009‐2010
Fourier series Dr.Eng Muhammad.A.R.Yass
Sultan
M O H D _ Y A S S 9 7 @ Y A H O O . C O M
2
Advance Mathematics
Fourier series
3rd Class
Electromechanical Eng.
Dr. Eng
Muhammad. A. R. Yass
3
Fourier series
Definition
The series
12 cos
∞
sin
Si the Fourier series of f on ( -L , L ) when the constant are chosen to be the Fourier coefficient of “f “on (-L , L)
Where
1 cos –
0 , 1 , 2
And
1 sin –
1 , 2 , 3
And
1
Example
Let f(x) = x for –π ≤ x ≤ π . we will write the Fourier series of “f” on [ -π , π] . the coefficient are
10
1cos
1cos sin 0
4
1 sin
1sin cos 0
2cos
21
Since cos ( n ) = (-1)n if n is an integer . the fourier series of “x” on [ - , ] is
2∞
1 sin 2 sin sin 223 sin 3
12 sin 4
25 sin 5 …
Example
Find the fourier series of the periodic function
1 0 10 1 2
Solution
d = 0
2 p = 2 p = 1
2 cos …… cos …… sin …… sin
1
11 1 0
| 1 1
1 cos
11
1 cos 0 cos
1sin | 0
1sin 0 0
f(t)
1 2
1
t
5
0
1 sin
1 1 sin 0 sin
1cos |
1cos
1cos 0
11 cos
1 1 cos 2/
12
1 cos 0
13 1 2
23
14 1 4 0
The fourier series become
12
2sin 0
23 sin 3 0
35 sin 5
Example
0 0
Solution
d = -
d + 2p =
p =
1
1
1
Always sin (n ) = 0
f(t)
t
-
6
1 cos–
1 –
cos cos
Integral by partial we get 0 if n = even and if n = odd
1 sin
1 sin
0
Fourier series will be
12
∞
cos 2 2 2
Example
Let f (x) = x for - ≤ x ≤
1
12 0
1cos
1cos
1cos sin 0
1cos
1cos
period
- 3 -3
7
1sin
1sin
sin cos
2cos
21
Fourier series will be
21 sin
∞
2 sin sin 223 sin 3
12 sin 4 ……
Example
Let
0 3 0 0 3
Solution
L = 3
13
13
32
13 cos 3
13 3
3cos 3 sin 3
31 1
13 sin 3
13 sin 3
3sin 3 cos 3
31
cos 1
period
3 -3
6
-6
8
The Fourier series
34
3∞
1 1 cos 33
1 sin 3
Example
Let
2 2 find fourier series
12 dx
18
12 cos
nπx2 dx
8 164
12 sin 2
3sin 3 cos 3
3 1
The Fourier series will be
34
3 1 1
∞
cos 33
1 sin 3
9
Even and odd function
Even Function
f is an even function on [-L , L ] if f(-x) = f(x) for -L ≤ x ≤ L
Odd Function
f is an odd function on [-L , L] if f(-x) = - f (x) -L≤ x ≤ L
Fig(1) Even function , cos
-3 +3
-1
Fig (2) Even function symmetrical a bout y-axis
Fig (3) Odd function , sin
Fig (4) Odd function symmetric through
the origin
+2 -2
+4
-4
+3 -3 -1 +1
-2 +2
+4
-1
10
If the function is even then
/
/
2
/
While if is odd then
/
/
0
Also even function
2
2cos 0
Then the function will be
2 cos
0 0
2 sin
sin ∞
Symmetric a bout the origin
11
Example
Find the fourier series of the function of the
2 1 01 0
2
Solution
Its an odd-function
0 0
2 sin
2 1 sin 2 sin
2 1cos
2cos
2 4cos
2cos 2
2 1 2 1
2cos 2
2
∞
1 2 1 cos 2 sin
Example
Find fourier series of the function
0 0
The function is odd then
0 ; 0
-
-2
+2
2 -
+1
-1
12
sin cos
1 10
∑ sin∞
Example
Let 1,1 find Fourier series
Solution
is an even function because f(-x) = f (x) ( i.e. example
f(-3) = f (3) on so on ) then 0
2 cos
2 cos8 6
1
the Fourier series
15 8
∞6 1 cos
Example
4 4
Solution
0
14 sin
4
12 sin 4 1 128 6/
-
h
-h
0
13
The Fourier series will be
1∞
128 6 sin 4
Conclusion
Even Function
Fourier series will be
12
∞
cos
2cos 0 , 1 , 2
Odd Function
Fourier series will be
∞
sin
2sin 1 , 2
Example
Find the Fourier series if f(x) = x2 0 < x < 2
Neither even nor odd
Period = 2L = 2 L =
0 2 -2
f(t)
4 4
14
1
83
1cos
1 sin 2
– cos2
sin
4 0
1sin
1
cos 2
cos 2
sin 4 0
1 sin
1
cos2
sin2cos 4
The Fourier series will be
43
4∞
cos 44
sin
#Example
“a” odd f (-x) = - f (x)
2 0 32 3 0
-2
2
3 6 -3 -6
15
“b” Neither EVEN nor odd
00 2 2
“C” EVEN f (-x) = f (x)
10 0 10 10
0
1
-1
2 -
-2
0 5 10 -10
2.5
16
Half Range Expansions
Half range fourier series if function f (x) is defined only in the half fourier interval (0 ) the equation of such function can be problem into other half of period (- 0) infinite way .
a) An odd b) An even c) Neither odd nor even
Example
Give f(x) = x in the interval 0 0 < x <
a) Find the Fourier series an a even function ( cos function) b) Find the Fourier series an odd function (sin function)
a- Even function bn = 0
2
2cos
U = x du = dx
dv = cos (nx) dx v = sin (nx)
2 sin
1sin
2 sin sin 0
1cos
2 1 cos cos 0
21 1
4
24 cos∞
, ,
17
b – odd function 0 0
2sin
sin 1cos
2 cos
1 cos 0
2 cos 0 cos 0
2 1
2 cos
21
2
2
2cos
2 cos∞
sin
Example
Find the sine and the cosine half range series of the function series
0
a. Even function
cos cos
cos 1
23
41
∞
cos
0
f(x)
18
b. Odd function
Solved Examples
The formula for a Fourier series on an interval [c,c+T] is:
Example (1)
Find the Fourier series for , .
Following the rules from the link above,
.
19
.
So,
Example (2)
Find the Fourier series for .
FOURIER SERIES BOOKS
20
Example (3)
Find the Fourier series for on
The general Fourier series on is:
The n = 0 case is not needed since the integrand in the formula for is .
In the present problem,
But since the right hand side is not defined if n = 0, the 0 index for a will have to be calculated seperately.
So the Fourier series is
for
21
Example (4) Find the Fourier series for
on
The general Fourier series on is:
In the present problem,
22
So the Fourier series is:
Setting x = 0 gives
Example (5) Find the Fourier series for
on
So the Fourier series is:
23
Example (6) Find the Fourier series for
on
The general Fourier series on is:
In the present problem,
So the Fourier series is:
on
Example (7) Find the Fourier series for a function
on .
Make the change of variables .
Now, look for the Fourier series of the function on
24
Since ,
Example (8)
Find the Fourier series for on .
A general formula for the Fourier series of a function on an interval is:
In the current problem, and .
The function is odd, so the cosine coefficients will all equal zero. Nevertheless, should still be calculated separately.
25
So the Fourier series for is
Example (9) Find the Fourier series for
This is the general Fourier Series:
So the given function can be replaced by its Fourier expansion:
26
So the solution is
27
Home Work
Problem (1) Find the Fourier series of the function
Answer.
Problem (2) . Find the Fourier series of the function
Answer. We have Therefore, the Fourier series of f(x) is
Problem (3) Find the Fourier series of
Answer.