Environmental Engineering Reference
In-Depth Information
5. Approximate Formulas
When
x
is sufficiently large, we have
2
π
2
π
x
cos
x
x
sin
x
−
4
−
2
π
−
4
−
2
π
J
γ
(
x
)
≈
,
Y
γ
(
x
)
≈
,
2
π
2
π
x
e
i
x
−
4
−
2
π
x
e
−
i
x
−
4
−
2
π
H
(
1
)
H
(
2
)
(
x
)
≈
,
(
x
)
≈
.
γ
γ
A.4 Legendre Polynomials
Consider the Legendre equation
1
x
2
y
−
2
xy
+
−
(
+
)
=
,
n
n
1
y
0
(A.19)
where
n
is a real constant. Its series solution takes the form
x
c
+
∞
+
∞
k
=
0
a
k
x
c
+
k
k
=
0
a
k
x
k
y
=
=
,
where
a
0
=
0. Substituting it into Eq. (A.19) leads to
+
∞
k
=
0
[(
k
+
c
)(
k
+
c
+
1
)
−
n
(
n
+
1
)]
a
k
x
k
+
c
−
+
∞
k
=
0
(
k
+
c
)(
k
+
c
−
1
)
a
k
x
k
+
c
−
2
+
=
0
.
(A.20)
Therefore all the coefficients must be zero. From the coefficients of
x
c
−
2
and
x
c
−
1
,
we obtain
c
(
c
−
1
)
a
0
=
0
,
c
(
c
+
1
)
a
1
=
0
.
Since
a
0
=
0,
c
=
0or
c
=
1, Eq. (A.20) can also be rewritten as
+
∞
k
=
0
[(
k
+
c
)(
k
+
c
+
1
)
−
n
(
n
+
1
)]
a
k
x
k
+
c
−
+
∞
∑
k
=
−
a
k
+
2
x
k
+
c
+
2
(
k
+
c
+
2
)(
k
+
c
+
1
)
=
0
.
By the coefficient of the general term, we have
a
k
+
2
=
(
k
+
c
)(
k
+
c
+
1
)
−
n
(
n
+
1
)
a
k
,
k
=
0
,
1
,
2
, ··· .
(A.21)
(
k
+
c
+
1
)(
k
+
c
+
2
)
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