Environmental Engineering Reference
In-Depth Information
A comparison of real and imaginary parts yields
cos
(
x
sin
θ
)=
J
0
(
x
)+
2
[
J
2
(
x
)
cos2
θ
+
J
4
(
x
)
cos4
θ
+
···
]
,
sin
(
x
sin
θ
)=
2
[
J
1
(
x
)
sin
θ
+
J
3
(
x
)
sin3
θ
+
···
]
.
Therefore, by the theory of Fourier series,
J
n
(
π
x
)
,
n
=
0
,
2
,
4
, ··· ,
1
π
cos
(
x
sin
θ
)
cos
n
θ
d
θ
=
0
,
n
=
1
,
3
,
5
, ··· ,
0
J
n
(
π
x
)
,
n
=
1
,
3
,
5
, ··· ,
1
π
sin
(
x
sin
θ
)
sin
n
θ
d
θ
=
0
,
n
=
0
,
2
,
4
, ··· .
0
Finally, we obtain an integral formula of Bessel functions by adding these two equa-
tions,
π
1
π
J
n
(
x
)=
cos
(
n
θ
−
x
sin
θ
)
d
θ
,
n
=
0
,
1
,
2
, ··· .
0
Similarity with Sine and Cosine Functions
A comparison of power series expansions of cos
x
and
J
0
(
x
)
shows that
J
0
(
x
)
is
quite similar to cos
x
. Similarly,
J
1
(
x
)
is similar to sin
x
. For example,
J
0
(
x
)
is a
even function, while
J
1
(
0 has no complex root but
an infinite number of distinct real roots. The approximate values of its positive real
roots are,
x
)
is an odd function.
J
0
(
x
)=
2
.
405
,
5
.
520
,
8
.
654
,
11
.
792
,
···
0 has no complex root but an infinite number of real roots. The
approximate values of its positive real roots are
Similarly,
J
1
(
x
)=
3
.
832
,
7
.
061
,
10
.
173
,
13
.
324
,
···
Thus, the zero points of
J
0
(
x
)
and
J
1
(
x
)
occur alternately. Also, the distance between
two adjoining zero points of
J
0
(
x
)
or
J
1
(
x
)
tends to
π
as
|
x
|→
+
∞
.
J
0
(
x
)
and
J
1
(
x
)
are thus called
periodic functions with a period of almost
2
π
. The graphs of
J
0
(
x
)
and
J
1
(
are also quite similar to those of cos
x
and sin
x
.
Similarly, we have the following properties for
J
n
(
x
)
x
)
with
n
as an integer: (1)
J
n
(
has no complex zero-points, but an infinite number of real zero-points. All
zero points are symmetrically distributed with respect to
x
x
)
=
0. All zero-points ex-
cept
x
=
0 are single zero-points; (2) the zero-points of
J
n
(
x
)
and
J
n
+
1
(
x
)
occur
μ
(
n
)
m
+
1
−
μ
(
n
m
tends to
μ
(
n
m
stands for the
m
-
alternately; (3) the
π
as
m
→
+
∞
.Here
th positive zero-point of
J
n
(
x
)
. Therefore,
J
n
(
x
)
is a
periodic function with a period
of almost
2
π
.
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