Environmental Engineering Reference
In-Depth Information
where
v
0
is the amplitude of plane waves and e
i
kr
cos θ
is their amplitude factor.
Therefore Eq. (A.32) shows that plane waves can be expanded by those of cylindri-
cal waves.
We may also explain the physical meaning of the generating function of Legendre
polynomials. Consider two particles of unit mass located at the origin
O
and at a
point
P
of distance
R
from the origin. When the particle at point
P
is moved infinitely
far away, the work done by gravitation is
+
∞
+
∞
R
=
1
R
.
This is called the
Newton potential
at point
P
that is due to the particle at the origin.
The Newton potential at point
P
due to the particle of unit mass at point
P
is thus
1
R
=
1
r
2
d
r
1
r
W
=
=
−
R
1
1
√
r
2
θ
,
where
r
and
r
are the position vectors of
P
and
P
respectively,
r
|
=
|
r
−
r
2
2
rr
cos
+
−
r
|
=
r
and
|
r
|
=
r
,
|
r
r
,
x
. Thus
1
1
is the angle between
r
and
r
.Let
t
r
√
1
θ
=
=
cos
θ
R
=
t
2
.
−
2
xt
+
1, in particular, we have
1
1
t
2
. Therefore, the generating
function of the Legendre poly-nomial represents the Newton potential.
When
r
=
R
=
√
1
−
2
xt
+
Remark 3.
The Euler integral of the first kind is a generalized integral containing
two positive parameters
p
and
q
,
1
x
p
−
1
q
−
1
d
x
B
(
p
,
q
)=
(
1
−
x
)
,
p
>
0
,
q
>
0
,
(A.34)
0
which is called the
Beta function
,the
B-function
for short, and is continuous when
p
>
0and
q
>
0
.
By a variable transformation
x
=
1
−
t
,wemayshowthesymmetryofthe
B
−
function,
i.e.
B
(
p
,
q
)=
B
(
q
,
p
)
.
We may also obtain the recurrence formula by using the integration by parts for
p
>
0and
q
>
1
1
0
(
1
p
q
−
1
d
x
p
B
(
p
,
q
)=
1
−
x
)
1
q
−
1
x
p
q
−
2
d
x
=
(
1
−
x
)
p
0
q
−
1
q
−
1
=
B
(
p
,
q
−
1
)
−
B
(
p
,
q
)
p
p
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