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