Environmental Engineering Reference
In-Depth Information
Tabl e A. 1
Associated legendre polynomials of degree
n
and order
m
m0 1
2
3
4
l
0
P
0
(
x
)
P
1
(
1
P
1
(
x
)
)
=(
1
−
x
2
x
1
2
)
=
sin θ
P
2
(
x
)
=
P
2
(
x
)
=
3
(
1
−
x
2
2
P
2
(
x
)
)
2
x
x
2
3
(
1
−
)
3
2
(
3
2
sin2θ
=
1
−
cos 2
θ
)
=
P
3
(
P
3
(
P
3
(
3
P
3
(
x
)
x
)
x
)
x
)
x
2
3
2
(
=
15
(
1
−
)
x
2
x
2
2
=
15
(
1
−
)
x
2
5
x
2
=
1
−
)
(
−
1
)
15
4
(
15
4
(
3sinθ
=
cos
θ
−
cos3
θ
)
3
8
(
=
=
θ
+
θ
)
sin
5sin3
−
sin3
θ
)
P
4
(
x
)
=
P
4
(
x
)
=
P
4
(
x
)
=
P
4
(
x
)
4
P
4
(
x
)
5
2
(
1
−
x
2
15
2
(
1
−
x
2
2
x
x
2
1
2
105
(
1
−
)
(
7
x
3
−
3
x
2
)(
7
x
2
)
)
−
1
)
105
8
(
5
16
(
15
16
(
=
2sin2
θ
=
2sin2
θ
+
7sin4
θ
)
=
3
+
4cos2
θ
−
sin4
θ
)
−
7cos4
θ
)
For a natural number
n
,
Γ
(
n
+
1
)=
n
!. Also
1
2
=
√
π
.
Γ
We may show, by the ratio test, that the convergence radius of the Bessel function
J
γ
(
x
.
There is an important relation between the two Euler integrals
)(
γ
>
0
)
is
R
=+
∞
)=
Γ
(
p
)
Γ
(
q
)
B
(
p
,
q
,
p
>
0
,
q
>
0
.
(A.40)
Γ
(
p
+
q
)
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