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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