Definition 1 (Periodic functions)
A function f(t) is said to have a period T or to be periodic with period T if for all t,
f(t+T)=f(t), where T is a positive constant. The least value of T>0 is called the principal period or the fundamental period
or simply the period of f(t).
Example 1
The function
has periods
,
since
all equal
.
Example 2
Let
.
If f(x) has the period
then
has the period
T.
(substitute
)
Example 3
If
f has the period
T then
Definition 2 (Periodic expansion)
Let a function
f be declared on the interval [0,
T). The
periodic expansion
of
f is
defined by the formula
Definition 3 (Piecewise continuous functions)
A function f defined on
I=[
a,
b] is said to be
piecewise continuous on
I if and only if
 (i)
 there is a subdivision
such that f is continuous on each subinterval
and
 (ii)
 at each of the subdivision points
both onesided limits of
f exist.
Theorem 1
Let
f be continuous on
.
Suppose that the series

(1) 
converges uniformly to
f for all
.
Then

(2) 
Definition 4 (Fourier coefficients, Fourier series)
The numbers
a_{n} and
b_{n} are called the
Fourier coefficients of
f.
When
a_{n} and
b_{n} are given by (
2), the trigonometric series (
1)
is called the
Fourier series of the function
f.
Example 4
Let
f(
x) be defined in the interval [0,
T] and determined outside of this interval
by its periodic extension, i.e. assume that
f(
x) has the period
T. The Fourier series
corresponding to
f(
x) (with
)
is

(5) 
where the Fourier coefficients
a_{n} and
b_{n} are

(6) 

(7) 
Example 5
Let
a_{n} and
b_{n} be the Fourier coefficients of
f, see above.
The
phase angle form of the Fourier series of
f is
with
amplitudes
and
phase angles
See the
applet to gain intuition as to the effects of the amplitude and the phases on the function behavior.
Example 6
We compute the Fourier series of the function
f given by
Since
f is an odd function, so is
,
and therefore
a_{0}=0
For
the coefficient
b_{n} is given by
It follows
The Gibbs Phenomenon
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