Fourier Series - HP 50g User Manual

Examples of the plots generated by these functions, for Uo = 1, a = 2, b = 3, c
= 4, horizontal range = (0,5), and vertical range = (-1, 1.5), are shown in the
figures below:

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 ) = sin x, and cos(x+2 ) = cos x,
the functions sin and cos are 2 -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
f
where the coefficients a
1
a
0
T
f(t) = U
o
⎛
∑
) ( t a
⎜
0
⎝
n 1
and b
n n
T 2 /
∫
) ( ,
f t dt
T 2 /
T
∫
b
n
T
[1-(t-a)/(b-1)] [H(t-a)-H(t-b)].
2 n
a cos t
n
T
are given by
2
T 2 /
∫
a
n
T 2 /
T
2
2 / n
) ( sin
f t
2 /
T
2 n
⎞
b sin t
⎟
n
T
⎠
2
n
) ( cos
f t t
T
.
t dt
,
dt
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