Geology Reference
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being described are vector fields in spherical co-ordinate systems and the basis
functions are Legendre functions of the first kind.
We will make extensive use of the associated Legendre functions
P
n
(
x
)defined
in terms of the Legendre polynomials
P
n
(
x
)by
1)
m
1
x
2
m
/2
d
m
P
n
(
x
)
P
n
(
x
)
=
(
−
−
dx
m
.
(1.177)
1)
m
is omitted,
as is commonly done in the USA (Dahlen and Tromp, 1998). In the classical liter-
ature, Ferrers used the notation
T
n
(
x
) to distinguish this case. In a spherical polar
co-ordinate system (
r
,θ,φ), the angular dependence is conveniently described by
series of the spherical harmonic functions
This is Hobson's definition (Copson, 1955). Sometimes the factor (
−
P
n
(cosθ)
e
im
φ
(1.178)
for
m
≥
0. For all
m
,wetake
1)
m
(
n
−
m
)!
P
−
m
m
)!
P
n
.
=
(
−
(1.179)
n
(
n
+
Spherical harmonics are orthogonal under integration over a sphere. Thus,
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
,
(1.180)
+
0
0
n
m
where δ
−
k
is a product of Kronecker deltas. We can then expand an arbitrary,
time dependent, function on a sphere
f
(
r
,θ,φ;
t
), as
l
δ
∞
n
q
n
(
r
,
t
)
P
n
(cosθ)
e
im
φ
.
f
(
r
,θ,φ;
t
)
=
(1.181)
n
=
0
m
=−
n
Multiplying both sides of (1.181) by
P
l
(cosθ)
e
ik
φ
, integrating over the unit sphere
and using the orthogonality relation (1.180), yields
2π
π
1)
m
2
n
+
1
q
n
(
r
,
t
)
f
(
r
,θ,φ;
t
)
P
−
n
(cosθ)
e
−
im
φ
sinθ
d
θ
d
φ (1.182)
=
(
−
4π
0
0
cients
q
n
(
r
,
t
). If the function
f
(
r
,θ,φ;
t
)is
as the expression for the radial coe
real, then
1)
m
(
n
+
m
)!
q
−
m
(
n
−
m
)!
q
m
n
,
=
(
−
(1.183)
n
where the asterisk denotes the complex conjugate.
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