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ECE 6382 ECE 6382
Legendre Functions
Wave Equation in Spherical Coordinates Wave Equation in Spherical Coordinates
2 2
22 2
2 2 2 2 2
( , , )
0, 2 /
1 1 1sin 0
sin sin
( , , ) ( ) ( )
r
k k
r kr r r r r
r R r
Source - free scalar wave equation in spherical coordinates :
Assume . Then separation of varia
22
2
2
2
( 1) 0
1sin ( 1) 0
sin sin
( ) ( ) 0
d dRr kr n n R
dr dr
d d mn n
d d
m
bles
yields
(spherical Bessel's equation)
(Legendre's equation)
(Harmonic equation)
Solution of the Spherical Solution of the Spherical
Wave EquationWave Equation
22
(1) (2)
2
2
( 1) 0 ( ) ( )
( ) ( ), ( ), ( ), ( ),
1sin ( 1) 0 ( ) (cos ),
sin sin
(cos ) (cos ), (cos ) ,
n
n n n n n
m
n
m m m
n n n
d dRr kr n n R R r z kr
dr dr
z kr j kr n kr h kr h kr
d d mn n L
d d
L P Q n
where
where an
2
0
( ) ( ) 0 ( ) , 0 2
( , , ) ( ) ( ) ( ) ( ) (cos )
( , , ) (
im
m im
n n
mn n
m e m
r R z k L e
r a z k
integer if
an integer if
A superposition of these solutions over the allowable separation constants is
0
) (cos )m im
n
m n
L e
Legendre’s Equation Legendre’s Equation
2 22
2 2
22
2
cos
1 2 1 01
1 1 01
0,
x
d y dy mx x n n ydx dx x
d dy mx n n y
dx dx x
m
The separated equation in the angular variable :
or
(self - adjoint form)
For this is simply ;
for
Legendre's equation
0,m it is the associated Legendre's equation
Series Solution of Legendre’s Series Solution of Legendre’s
EquationEquation
2 2
0
2 2 2
2
2 12
1 2 2 !
2 ! 2 ! !
, :
, =
,
1 1ln ( )
2 1
n k n k
n nk
n n n
n
n
n
n n
n k xP x
k n k n k
n
n
xQ x P x n
x
where the "floor" of , is the largest integer contained in
even
odd
A second solution is given by
2
21
1 ! 1( )
2! !
1 1 1 ( ) 1 . 1.
2 3
m mn kn
m
n
n m x xm
m n m
n P x xn
where Only is finite at
Low Order Legendre FunctionsLow Order Legendre Functions
2 2
2
cos cos
1 1 10 : 1 1 ln ln cot2 1 2 2
11: cos ln 1 cos ln cot 1
2 1 2
1 1 3 1 1 3 3cos 1 3cos3 1 3cos 2 1 ln ln cot2 :
2 4 4 1 2 2 2 2
n n n nP x P Q x Q
xnx
x xn x
x
x x xxn
x
( ) ( ), ( )n n nL x P x Q x
Generating FunctionGenerating Function
2
0
,
1, , 1
1 2
n
n
n
g x t
g x t P x t txt t
The (integer order) Legendre functions can also be defined through a
:
The generating function definition also leads to the integral
representati
generating function
1
2
1
2 1 2
n
n
C
tP x dt
i xt t
on
Recurrence RelationsRecurrence Relations
20
2
1
22
32
1,
1 2
1,
1 2
, ( )1 21 2
n
n
n
n
n
n
g x t P x txt t
g x tt t xt t
x t x tg x t n P x t
xt txt t
Several recurrence relations are easily derived from the generating
function. For example, from ,
we note that
1 1(2 1) ( ) ( 1) ( ) ( )n n n
t
n xP x n P x nP x
Rearranging and equating like powers of yields
Recurrence Relations (cont.)Recurrence Relations (cont.)
1 2
1
1 1
2
1
1 1
2
1 1
( ) ( ) ( )
( ) 2 1 ( ) 1 ( )
( ) ( ) ( )
( ) ( ) ( )
1 ( ) ( ) ( )
2 1 ( ) ( ) ( )
1 ( ) ( )
:
) (
n n n
n n n
n n n
n n n
n n n
n n n
n n n
L x P x Q x
nL x n xL x n L x
xL x L x nL x
L x xL x nL x
x L x nL x nxL x
n L x L x L x
x L x nxL x nL x
For or
WronskiansWronskians
2 1
( ) ( )[ , ; ] ( ) ( )
(1 )(1 )
(1 )
dQ x dP xW P Q x P x Q x
dx dx
x
Plots of Legendre FunctionsPlots of Legendre Functions
P0(x)
Special Values and Relations Special Values and Relations
2
1 1, 1
1 1 , 1
1
1 1
11
2 !
n n
n
n n
n
n n
n
nn
n n n
P Q
P Q
P x P x
P x n n
x
dP x x
n dx
is an th degree polynomial with zeros
in
Rodrigue's formula
:
Solutions of the Associated Legendre Solutions of the Associated Legendre
EquationEquation
0 0
/2 /22 2
, ,
1 1 , 1 1
0, , ,
n n n n
m mm mm mn nm m
n nm m
m m
n n
P x P x Q x Q x
d P x d Q xP x x Q x x
dx dx
P x Q x m n m n
integers
Recurrence Relations for Associated Recurrence Relations for Associated
Legendre FunctionsLegendre Functions
12
1 1
1 1
2
12
12
2
= ,
1 2 1 0
21 0
1
1
1
11 1
1
1
m m m
n n n
m m m
n n n
m m m
n n n
m m m
n n n
m m m
n n n
m m
n n
L P Q
m n L n xL m n L n
mxL L m n n m L m
x
L nxL m n Lx
L n xL n m Lx
mxL L
x
For or
(recursion on )
(recursion on )
12
12
1
2
1
22
1
1
1
1 1
m
n
m m m
n n n
n m n mL
x
mxL L L
x x
Special Values of the Associated Special Values of the Associated
Legendre Functions Legendre Functions
/2 /2
0
0
1, 0(1) , (1) ,
0, 0
1 2 ! , even(0) ,2
0, odd
( ) 1 (0)
( ) 1 (0)
m m
n n
n m n m
m
n
rrm m r
n nr
x
rrm m r
n nr
x
mP Q
m
n mn m
P
n m
dP x P
dx
dQ x Q
dx
Orthogonalities Orthogonalities
1
1
0
1
2
1
0
2
2
0
!2cos cos sin
2 1 !
1
cos cos !1
sin !
cos coscos cos sin
sin
!2( 1)
2 1 !
m m
p q
m m
p q pq
m n
p p
m n
p p
mn
m m
p q m m
p q
pq
P x P x dx
q mP P d
q q m
P x P xdx
x
P P p md
m p m
dP dP mP P d
d d
p mp p
p p m
The Tesseral Harmonics and Their The Tesseral Harmonics and Their
Orthogonalities Orthogonalities
2
0 0
!2 1, cos
4 !
, , sin
m m im
n n
m n
p q pq mn
n mnY P e
n m
Y Y d d
The ( ) :
Orthogonalities of the tesseral harmonics :
tesseral spherical harmonics
Tesseral Harmonic ExpansionTesseral Harmonic Expansion
0
2
0 0
( , ) ,
( , ) , sin
!2 1, cos
4 !
nm
mn n
n m n
m
mn n
m m im
n n
f a Y
a f Y d d
n mnY P e
n m
Expansion of a function on a spherical surface :
where
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