Geology Reference
In-Depth Information
They are orthogonal under integration over a sphere, or
2π
π
1)
m
4
π
2
n
P
n
(cosθ)
P
l
(cosθ)sinθ
e
i
(
m
+
k
)φ
d
θ
d
φ
=
n
m
(
−
1
δ
l
δ
−
k
.
(B.7)
+
0
0
For two points with spherical polar co-ordinates (θ,φ)and(θ
,φ
) on a sphere of
arbitrary radius, by the law of cosines for spherical triangles,
cosθcosθ
+
sinθsinθ
cos(φ
−
φ
),
Θ=
cos
(B.8)
with
the angle between the radius vectors through the two points. The addition
theorem for spherical harmonics (Sommerfeld, 1964, pp. 133-144) gives
Θ
n
1)
m
P
n
(cosθ)
P
−
n
(cosθ
)
e
im
(φ
−
φ
)
P
n
(cos
Θ
)
=
(
−
.
(B.9)
m
=−
n
Associated Legendre functions of the first kind have the closed-form expression,
m
)!
1
z
2
m
/2
z
n
−
m
,
(2
n
)!
2
n
n
! (
n
P
n
(
z
)
1)
m
c
2
z
n
−
m
−
2
=
(
−
−
+
+···
(B.10)
−
where
z
n
−
m
is an even polynomial in
z
for
n
c
2
z
n
−
m
−
2
+
+···
−
m
even, an odd
polynomial in
z
for
n
m
odd. Associated Legendre functions of the second kind
have logarithmic singularities at
−
1 on the real axis and are denoted by
Q
n
(cosθ).
The associated Legendre functions of the first kind have properties that we will
use in many contexts, while the associated Legendre functions of the second kind
will find only rare application in this work.
±
B.1 Recurrence relations
Among the most important properties of the Legendre functions of the first and
second kinds are the recurrence relations that both obey. We record them here for
functions of cosθ
=
z
of the first kind, with degree
n
and azimuthal number
m
,
denoted by
P
n
(cosθ).
The recurrence relations for fixed azimuthal number but variable degree are
1
(
n
m
)
P
n
−
1
,
1
2
n
+
cosθ
P
n
=
1)
P
n
+
1
+
−
m
+
(
n
+
(B.11)
1
n
(
n
m
)
P
n
−
1
.
dP
n
d
θ
=
1
1)
P
n
+
1
−
sinθ
−
m
+
(
n
+
1)(
n
+
(B.12)
2
n
+
The recurrence relations for fixed degree but variable azimuthal number are
dP
n
d
θ
=
2
P
m
+
1
m
)
P
m
−
n
,
1
−
(
n
−
m
+
1)(
n
+
(B.13)
n
P
m
+
1
1)
P
m
−
n
.
sin
θ
2
m
cosθ
P
n
=−
+
(
n
+
m
)(
n
−
m
+
(B.14)
n
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