13. Consider the Fourier series of f(x)
5 - x, 0<x<
x-<x< 1
The Fourier coefficient by is
b)
c)2
d) 1
Answers
Answer:
Baron Jean Baptiste Joseph Fourier
(
1768
−
1830
)
introduced the idea that any periodic function can be represented by a series of sines and cosines which are harmonically related.
Baron Jean Baptiste Joseph Fourier (1768−1830)
Fig.1 Baron Jean Baptiste Joseph Fourier (1768−1830)
To consider this idea in more detail, we need to introduce some definitions and common terms.
Basic Definitions
A function
f
(
x
)
is said to have period
P
if
f
(
x
+
P
)
=
f
(
x
)
for all
x
.
Let the function
f
(
x
)
has period
2
π
.
In this case, it is enough to consider behavior of the function on the interval
[
−
π
,
π
]
.
Suppose that the function
f
(
x
)
with period
2
π
is absolutely integrable on
[
−
π
,
π
]
so that the following so-called Dirichlet integral is finite:
π
∫
−
π
|
f
(
x
)
|
d
x
<
∞
;
Suppose also that the function
f
(
x
)
is a single valued, piecewise continuous (must have a finite number of jump discontinuities), and piecewise monotonic (must have a finite number of maxima and minima).
If the conditions
1
and
2
are satisfied, the Fourier series for the function
f
(
x
)
exists and converges to the given function (see also the Convergence of Fourier Series page about convergence conditions.)
At a discontinuity
x
0
, the Fourier Series converges to
lim
ε
→
0
1
2
[
f
(
x
0
−
ε
)
−
f
(
x
0
+
ε
)
]
.
The Fourier series of the function
f
(
x
)
is given by
f
(
x
)
=
a
0
2
+
∞
∑
n
=
1
{
a
n
cos
n
x
+
b
n
sin
n
x
}
,
where the Fourier coefficients
a
0
,
a
n
,
and
b
n
are defined by the integrals
a
0
=
1
π
π
∫
−
π
f
(
x
)
d
x
,
a
n
=
1
π
π
∫
−
π
f
(
x
)
cos
n
x
d
x
,
b
n
=
1
π
π
∫
−
π
f
(
x
)
sin
n
x
d
x
.
Sometimes alternative forms of the Fourier series are used. Replacing
a
n
and
b
n
by the new variables
d
n
and
φ
n
or
d
n
and
θ
n
,
where
d
n
=
√
a
2
n
+
b
2
n
,
tan
φ
n
=
a
n
b
n
,
tan
θ
n
=
b
n
a
n
,
we can write:
f
(
x
)
=
a
0
2
+
∞
∑
n
=
1
d
n
sin
(
n
x
+
φ
n
)
or
f
(
x
)
=
a
0
2
+
∞
∑
n
=
1
d
n
cos
(
n
x
+
θ
n
)
.
Fourier Series of Even and Odd Functions
The Fourier series expansion of an even function
f
(
x
)
with the period of
2
π
does not involve the terms with sines and has the form:
f
(
x
)
=
a
0
2
+
∞
∑
n
=
1
a
n
cos
n
x
,
where the Fourier coefficients are given by the formulas
a
0
=
2
π
π
∫
0
f
(
x
)
d
x
,
a
n
=
2
π
π
∫
0
f
(
x
)
cos
n
x
d
x
.
Accordingly, the Fourier series expansion of an odd
2
π
-periodic function
f
(
x
)
consists of sine terms only and has the form:
f
(
x
)
=
∞
∑
n
=
1
b
n
sin
n
x
,
where the coefficients
b
n
are
b
n
=
2
π
π
∫
0
f
(
x
)
sin
n
x
d
x
.
Below we consider expansions of
2
π
-periodic functions into their Fourier series, assuming that these expansions exist and are convergent.