nn n p 1 () pn n n () , ,,, hh =-= 2 1 012 òò + nm n m mn ... · sphericalbessel functions 12 11...
TRANSCRIPT
Legendre polynomials
( )221 1 0 1 2
2( )
( ) , , , ,!
n n n
n n
dP n
n dh h
h
-= - =
1
1 0
22 1
( ) ( ) (cos ) (cos )sinn m n m mnP P d P P dn
p
h h h q q q q d-
= =+ò ò
Associated Legendre functions22 21 1 1 0 1
2( )
( ) ( ) ( ) , , , , ,!
n l n lll n
n n n l
dP l n
n dh h h
h
+ +
+
-= - - =
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
η
Pnl (
η)
P11
P10
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
η
Pnl (
η)
P20
P21
P22
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
η
Pnl (
η)
P30
P31
P32
P33
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
η
Pnl (
η)
P41
P42
P43
P44
P40
1
1
22 1( )!
( ) ( ) ,( )( )!
l ln n nn
n lP P d
n n lh h h d¢ ¢
-
+=
+ -ò1
21 1
( )!( ) ( ) .
( )!l ln n ll
d n lP P
l n lh
h h dh
¢¢
-
+=
--ò
Spherical harmonics2 1
4( )( )!
( , ) (cos ) ,( )!
l l iln n
n n lY P e
n ljq j q
p+ -
=+
{ } 2 14
( )( )!Re ( , ) (cos )cos ,
( )!l ln n
n n lY P l
n lq j q j
p+ -
=+
2
0 0
*( , ) ( , )sinl ln n nn llY Y d d
p p
q j q j q q j d d¢¢ ¢ ¢=ò ò
Spherical harmonics
Spherical Bessel functions
1 21 12 2
2 2
( , )( ) ( ), ( ) ( ), ( ) ( )n n n n nn n
j J y Y h j yp p
r r r r r rr r+ +
= = =
( ) ( )1 1 sin cos( ) , ( )
n nn n
n nd d
j yd d
r rr r r r
r r r r r r
æ ö æ ö÷ ÷ç ç= - = - -÷ ÷ç ç÷ ÷÷ ÷ç çè ø è ø
0 0
1 12 2
2 22 2
3 3 3 31 1
sin cos( ) , ( ) ,
sin cos cos sin( ) , ( ) ,
sin cos cos sin( ) , ( ) ,
j y
j y
j y
r rr r
r rr r r r
r rr rr r
r r r rr r
r r r rr r
= = -
= - = - -
æ ö æ ö÷ ÷ç ç÷ ÷= - - = - + -ç ç÷ ÷ç ç÷ ÷ç çè ø è ø