404 Chapter 14Partial differential equations
The Θequation
(14.60)
The equation is transformed into the associated Legendre equation (13.22) by means
of the substitutionx 1 = 1 cos 1 θ. We have
Therefore,
and equation (14.60) becomes
(14.65)
and this is identical to the associated Legendre equation. The finite solutions in the
interval − 11 ≤ 1 x 1 ≤ 11 ( 01 ≤ 1 θ 1 ≤ 1 π) are the (normalized) associated Legendre functions
(equation (13.28))
(14.66)
These functions form an orthormal set, with property (equation (13.29))
(14.67)
Spherical harmonics
The products of the angular functionsΘ
l,mandΦ
m,
(14.68)
Yllm
lm,,lm m,= =
−!
() ()()
(||)
θθφφΘΦ
21
4 π ((||)
(cos )
||lm
Pe
lmim+!
θ
φZ
02 πΘΘ
lm l m lld
, ′ ,′(cos ) (cos ) sin =
,θθθθδ
Θ
lm lmllm
lm
P
,=
+−!
+!
(cos )
()(||)
(||)
(cos
||θ
21
2
θθ)
,, ,,
,, ,,
,
=
=±± ±
l
ml
0123
012
...
...
() ()
()
121
1
22222−−++−
−
x
d
dx
x
d
dx
ll
m
x
ΘΘ
ΘΘ= 0
1
22sin
sin
cos
θθ sin
θ
θ
θ
θ
θθ
d
d
d
d
d
d
d
d
ΘΘ Θ
=+ =(()12
222−−x
d
dx
x
d
dx
ΘΘ
d
d
d
dx
d
dx
x
d
dx
x
222222221
ΘΘΘ Θ
θ
=−cosθθ+sin = −( ) −
dd
dx
Θ
d
d
d
dx
dx
d
d
dx
ΘΘ Θ
θθ
==−sinθ
1
1
22sin
sin ( )
sin
θθ
θ
θ
θd
d
d
d
ll
Θ m
++−
Θ= 0