Appendix A Recursions of associated Legendre functions€¦ · Appendix A Recursions of associated...
Transcript of Appendix A Recursions of associated Legendre functions€¦ · Appendix A Recursions of associated...
Appendix A Recursions of associated Legendre functions
!
Pn
m :
!
P0
0(x) =1 (1)
!
Pn
n(x) = (2n "1) 1" x
2Pn"1
n"1(x) ...
!
"
!
n "1 (2)
!
Pn
n"1(x) = (2n "1)xP
n"1
n"1(x) ...
!
"
!
n "1 (3)
!
Pn
m(x) = 1
n"m(2n "1)xP
n"1
m(x) " (n + m "1)P
n"2
m(x)( ) ...
!
"
!
m " 0
!
"
!
n " 2 # m (4)
!
n = 0,1,2,3,K
!
m = 0,K,+n
!
x = sin"
!
" ... elevation angle;
!
P00(sin" ) =1
!
P10(sin" ) = sin"
!
P11(sin" ) = #cos"
!
P20(sin" ) = 1
2(3cos
2" #1)
!
P21(sin" ) = #3cos" sin"
!
P22(sin" ) = 3cos
2"
!
P30(sin" ) = 1
2sin" (5sin2" # 3)
!
P31(sin" ) = # 3
2cos" (5sin2" #1)
!
P32(sin" ) =15cos
2" sin" Condon-Shortley transformed associated Legendre functions
!
˜ P n
m :
!
˜ P n
|m |(sin" ) = (#1)
mP
n
|m |(sin" ) (5)
!
n = 0,1,2,3,K
!
m = 0,K,+n
!
˜ P 00(sin" ) =1
!
˜ P 10(sin" ) = sin"
!
˜ P 11(sin" ) = cos"
!
˜ P 20(sin" ) = 1
2(3cos
2" #1)
!
˜ P 21(sin" ) = 3cos" sin"
!
˜ P 22(sin" ) = 3cos
2"
!
˜ P 30(sin" ) = 1
2sin" (5sin
2" # 3)
!
˜ P 31(sin" ) = 3
2cos" (5sin
2" #1)
!
˜ P 32(sin" ) =15cos
2" sin"
!
˜ P 33(sin" ) =15cos
3"
orthonormalized complex spherical harmonics
!
Yn
m :
!
Yn
m(" ,#) =
2n +1
4$
(n% | m |)!
(n+ | m |)!˜ P
n
|m |(sin" )e
im# (6)
!
n = 0,1,2,3,K
!
m = "n,K,+n
!
" ... azimuth angle orthonormalized real spherical harmonics
!
ˆ Y n
m :
!
ˆ Y n
+m(" ,#) =
(2 $%m
)(2n +1)
4&
(n $m)!
(n + m)!˜ P
n
m(sin" )cos(m#) (7)
!
n = 0,1,2,3,K
!
m = 0,K,+n
!
ˆ Y n
"m(# ,$) =
(2 "%m
)(2n +1)
4&
(n "m)!
(n + m)!˜ P
n
m(sin# )sin(m$) (8)
!
n =1,2,3,K
!
m =1,K,+n
!
"m
... Kronecker delta
!
ˆ Y 00(sin" ) = 1
4#
!
ˆ Y 1"1
(sin# ) = 3
4$cos# sin%
!
ˆ Y 10(sin" ) = 3
4#sin"
!
ˆ Y 11(sin" ) = 3
4#cos" cos$
!
ˆ Y 2"2
(sin# ) = 15
16$cos
2# sin2%
!
ˆ Y 2"1
(sin# ) = 15
4$cos# sin# sin%
!
ˆ Y 20(sin" ) = 5
16#(3cos
2" $1)
!
ˆ Y 2+1
(sin" ) = 15
4#cos" sin" cos$
!
ˆ Y 2+2
(sin" ) = 15
16#cos
2" cos2$
!
ˆ Y 3"3
(sin# ) = 35
32$cos
3# sin3%
!
ˆ Y 3"2
(sin# ) = 105
16$cos
2# sin# sin2%
!
ˆ Y 3"1
(sin# ) = 21
32$cos# (5sin
2# "1)sin%
!
ˆ Y 30(sin" ) = 7
16#sin" (5sin
2" $ 3)
!
ˆ Y 3+1
(sin" ) = 21
32#cos" (5sin
2" $1)cos%
!
ˆ Y 3+2
(sin" ) = 105
16#cos
2" sin" cos2$
!
ˆ Y 3+3
(sin" ) = 35
32#cos
3" cos3$
semi-normalized real spherical harmonics
!
Y n
m :
!
Y n
m(" ,#) =
4$
2n +1ˆ Y n
m(" ,#) (9)
!
n = 0,1,2,3,K
!
m = "n,K,+n
!
Y 00(sin" ) =1
!
Y 1"1(sin# ) = cos# sin$
!
Y 10(sin" ) = sin"
!
Y 11(sin" ) = cos" cos#
!
Y 2"2(sin# ) = 3
4cos
2# sin2$
!
Y 2"1(sin# ) = 3cos# sin# sin$
!
Y 20(sin" ) = 1
2(3cos
2" #1)
!
Y 2+1(sin" ) = 3cos" sin" cos#
!
Y 2+2(sin" ) = 3
4cos
2" cos2#
!
Y 3"3(sin# ) = 5
8cos
3# sin3$
!
Y 3"2(sin# ) = 15
4cos
2# sin# sin2$
!
Y 3"1(sin# ) = 3
8cos# (5sin2# "1)sin$
!
Y 30(sin" ) = 1
2sin" (5sin2" # 3)
!
Y 3+1(sin" ) = 3
8cos" (5sin2" #1)cos$
!
Y 3+2(sin" ) = 15
4cos
2" sin" cos2#
!
Y 3+3(sin" ) = 5
8cos
3" cos3#
not-normalized real spherical harmonics = reference spherical harmonics
!
( Y
n
m :
!
( Y
n
m(" ,#) =
4$
2n +1
(n+ | m |)!
(n% | m |)!ˆ Y n
m(" ,#) (10)
!
n = 0,1,2,3,K
!
m = "n,K,+n
!
( Y 00(sin" ) =1
!
( Y 1"1(sin# ) = cos# sin$
!
( Y 10(sin" ) = sin"
!
( Y 11(sin" ) = cos" cos#
!
( Y 2"2(sin# ) = 3cos
2# sin2$
!
( Y 2"1(sin# ) = 3cos# sin# sin$
!
( Y 20(sin" ) = 1
2(3cos
2" #1)
!
( Y 2
+1(sin" ) = 3cos" sin" cos#
!
( Y 2
+2(sin" ) = 3cos
2" cos2#
!
( Y 3"3(sin# ) =15cos
3# sin3$
!
( Y 3"2(sin# ) =15cos
2# sin# sin2$
!
( Y 3"1(sin# ) = 3
2cos# (5sin2# "1)sin$
!
( Y 30(sin" ) = 1
2sin" (5sin2" # 3)
!
( Y 3
+1(sin" ) = 3
2cos" (5sin2" #1)cos$
!
( Y 3
+2(sin" ) =15cos
2" sin" cos2#
!
( Y 3
+3(sin" ) =15cos
3" cos3# orthonormalized weights
!
ONn
m related to reference spherical harmonics:
!
ON0
0= 1
4"; (11)
!
ON1
"1= 3
4#,
!
ON1
0= 3
4",
!
ON1
+1= 3
4";
!
ON2
"2= 5
48#,
!
ON2
"1= 5
12#,
!
ON2
0= 5
4",
!
ON2
+1= 5
12",
!
ON2
+2= 5
48";
!
ON3
"3= 7
1440#,
!
ON3
"2= 7
240#,
!
ON3
"1= 7
24#,
!
ON3
0= 7
4",
!
ON3
+1= 7
24",
!
ON3
+2= 7
240",
!
ON3
+3= 7
1440";
semi-normalized weights
!
SNn
m related to reference spherical harmonics:
!
SN0
0=1; (12)
!
SN1
"1=1,
!
SN1
0=1,
!
SN1
+1=1;
!
SN2
"2= 1
12,
!
SN2
"1= 1
3,
!
SN2
0=1,
!
SN2
+1= 1
3,
!
SN2
+2= 1
12;
!
SN3
"3= 1
360,
!
SN3
"2= 1
60,
!
SN3
"1= 1
6,
!
SN3
0=1,
!
SN3
+1= 1
6,
!
SN3
+2= 1
60,
!
SN3
+3= 1
360;