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Fourier Series for Three Periodic Functions

​
function
sawtooth 1 (even function)
sawtooth 2 (odd function)
square wave (neither even nor odd)
number of terms
1
2
3
4
5
6
Fourier coefficients
a
0
=
1
2
a
n
:
4
2
π
b
n
:
{0}
Periodic phenomena occur frequently in nature. Fourier series approximate periodic functions using trigonometric functions. This Demonstration shows three functions and their approximations using Fourier series. The functions are an even function,
g(x)=g(-x)
, an odd function,
h(x)=-h(x)
, and a function that is neither even nor odd.
For a function
f
defined on the interval
[-L,L]
, the Fourier coefficients are defined by:
a
0
=
1
L
L
∫
-L
f(x)dx
,
a
n
=
1
L
L
∫
-L
f(x)cos
nπx
L
dx
,
b
n
=
1
L
L
∫
-L
f(x)sin
nπx
L
dx
.
With those coefficients and for
f
suitably well behaved,
f(x)=
1
2
a
0
+
∞
∑
n=1
a
n
cos
nπx
L
+
b
n
sin
nπx
L
.
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