Geology Reference
In-Depth Information
Appendix B
Properties of Legendre functions
Legendre functions arise from the separation of variables for Laplace's equation in
spherical polar co-ordinates (
r
,θ,φ). The separated functional dependence on
r
for
integer
n
can be either
r
n
for internal sources or 1/
r
n
+
1
for external sources, while
that for φ is
e
im
φ
for integer
m
. The separated functional dependence onθ,sayw(θ),
obeys Legendre's equation,
1
sinθ
sinθ
n
(
n
m
2
sin
2
d
d
θ
d
d
θ
+
+
1)
−
w
=
0,
(B.1)
θ
where
n
is the integer degree and
m
is the integer order, with
|
m
|≤
n
. Legendre's
equation has regular singularities at infinity and at
1ontherealaxis.Legendre
functions of the first kind are regular in the finite plane and are denoted by
P
n
(cosθ). For
m
±
=
0, they are polynomials of degree
n
in
z
=
cosθ and can be
generated by Rodrigues' formula,
dz
n
z
2
1
n
d
n
1
2
n
n
!
P
n
(
z
)
=
−
,
(B.2)
giving
P
0
(
z
)
z
,etc.For
m
0, they are called associated Legendre
functions of the first kind, and for 0
=
1,
P
1
(
z
)
=
≤
m
≤
n
are generated, in turn, by
1)
m
1
z
2
m
/2
2
n
n
!
−
dz
n
+
m
z
2
1
n
d
n
+
m
P
n
(
z
)
=
(
−
−
(B.3)
1)
m
1
z
2
m
/2
d
m
P
n
(
z
)
=
(
−
−
dz
m
.
(B.4)
For
m
≤
0,
1)
m
(
n
−
m
)!
P
−
m
m
)!
P
n
=
(
−
(B.5)
n
(
n
+
with
|
m
|≤
n
. Spherical harmonic functions have the form
P
n
(cosθ)
e
im
φ
.
(B.6)
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