Environmental Engineering Reference
In-Depth Information
,
H
(
1
)
and
H
(
2
)
,they
also have corresponding differential properties. The differential properties of Bessel
functions can also be expressed in an integral form.
Since
Y
γ
(
x
)
(
x
)
(
x
)
are formed by linear combinations of
J
γ
(
x
)
γ
γ
Bessel Functions of Semi-Odd Order
Note that
1
2
3
2
+
m
=
√
π
,
√
π
,
1
·
3
·
5
····
(
2
m
+
1
)
Γ
Γ
=
m
=
0
,
1
,
2
, ··· .
2
m
+
1
Thus
2
π
x
2
+
∞
m
=
0
m
2
+
2
m
(
−
1
)
3
2
+
m
J
2
(
x
)=
=
x
sin
x
.
m
!
Γ
2
π
1
2
, the formula
2
(
)=
γ
=
−
Similarly,
J
x
x
cos
x
.For
n
−
2
x
γ
J
γ
+
1
(
x
)=
J
γ
(
x
)
−
J
γ
−
1
(
x
)
yields
n
J
n
−
2
x
1
2
J
n
+
2
(
x
)=
−
2
(
x
)
−
J
n
−
2
(
x
)
.
(A.17)
1
1
3
n
the formula
J
γ
−
1
(
1
2
2
x
γ
For
γ
−
1
=
−
+
x
)=
J
γ
(
x
)
−
J
γ
+
1
(
x
)
leads to
1
2
−
n
J
2
x
J
n
2
(
x
)=
n
2
(
x
)
−
J
2
(
x
)
.
(A.18)
1
1
3
−
+
−
−
−
n
+
A repeated application of Eqs. (A.17) and (A.18) thus yields
n
2
π
2
1
x
n
sin
x
x
d
d
x
1
x
n
+
J
n
+
2
(
x
)=(
−
1
)
,
1
2
π
2
1
x
n
cos
x
x
d
d
x
1
x
n
+
n
+
2
(
J
x
)=
,
1
−
where
1
x
n
d
d
x
1
x
d
d
x
stands for the
operation on a function
n
-times so that
1
x
n
d
n
d
x
n
.
Therefore we may obtain Bessel functions of semi-odd order by a finite number
of the fundamental arithmetic operations of sine, cosine and power functions. For
this reason they are called the elementary functions.
d
d
x
1
x
n
=
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