# Fourier Series - HP 49g+ User Manual

Graphing calculator.

## 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.
expanded into a series of sine and cosine functions known as a Fourier series
given by
f
) (
t
a
0
where the coefficients a
1
T
2 /
a
0
T
2 /
T
b
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,
2
that the function f(t) = t
+t is periodic with period T = 2. To determine the
coefficients a
, a
, and b
0
1
follows: First, define function f(t) = t
Next, we'll use the Equation Writer to calculate the coefficients:
A T-periodic function f(t) can be
2
n
π
a
cos
t
n
T
n
1
and b
are given by
n
n
2
T
2 /
) (
,
f
t
dt
a
n
T
2 /
T
2
T
2 /
n
) (
sin
f
t
n
T
2 /
T
for the corresponding Fourier series, we proceed as
1
2
+t :
2
n
π
b
sin
t
n
T
2
π
n
) (
cos
,
f
t
t
dt
T
π
.
t
dt
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