Page 16-27
Fourier series
Fourier series are series involving sine and cosine functions typically used to
expand periodic functions. A function f(x) is said to be periodic
, of period T,
if f(x+T) = f(t). For example, because sin(x+2π) = sin x, and cos(x+2π) = cos
x, the functions sin and cos are 2π-periodic functions. If two functions f(x) and
g(x) are periodic of period T, then their linear combination h(x) = a⋅f(x) +
b⋅g(x), is also periodic of period T. A T-periodic function f(t) can be
expanded into a series of sine and cosine functions known as a Fourier series
given by
∑
∞
=
⋅+⋅+=
1
0
2
sin
2
cos)(
n
nn
t
T
n
bt
T
n
aatf
ππ
where the coefficients a
n
and b
n
are given by
∫∫
−−
⋅⋅=⋅=
2/
2/
2/
2/
0
,
2
cos)(
2
,)(
1
T
T
T
T
n
dtt
n
tf
adttf
a
∫
−
⋅⋅=
2/
2/
.
2
sin)(
T
T
n
dtt
T
n
tfb
The following exercises are in ALG mode, with CAS mode set to Exact.
(When you produce a graph, the CAS mode will be reset to Approx. Make
sure to set it back to Exact after producing the graph.) Suppose, for example,
that the function f(t) = t
2
+t is periodic with period T = 2. To determine the
coefficients a
0
, a
1
, and b
1
for the corresponding Fourier series, we proceed as
follows: First, define function f(t) = t
2
+t :
Next, we’ll use the Equation Writer to calculate the coefficients:
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