HP 49g+ - Manual (Page 501)

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+2
π) = 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
n n
t
T
n
b t
T
n
a a t f
π π
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
dt t
T
n
t f
T
a dt t f
T
a
π
∫
−
⋅ ⋅ =
2 /
2 /
.
2
sin ) (
T
T
n
dt t
T
n
t f b
π
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: