Appendix A Recursions of associated Legendre functions€¦ · Appendix A Recursions of associated...

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Appendix A Recursions of associated Legendre functions P n m : P 0 0 ( x ) = 1 (1) P n n ( x ) = (2n " 1) 1 " x 2 P n "1 n "1 ( x ) ... " n " 1 (2) P n n"1 ( x ) = (2n " 1) xP n"1 n"1 ( x ) ... " n " 1 (3) P n m ( x ) = 1 n " m (2 n " 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; P 0 0 (sin " ) = 1 P 1 0 (sin " ) = sin" P 1 1 (sin" ) = # cos" P 2 0 (sin" ) = 1 2 (3cos 2 " # 1) P 2 1 (sin" ) = #3cos" sin" P 2 2 (sin" ) = 3cos 2 " P 3 0 (sin" ) = 1 2 sin" (5sin 2 " # 3) P 3 1 (sin" ) = # 3 2 cos" (5sin 2 " # 1) P 3 2 (sin" ) = 15cos 2 " sin" Condon-Shortley transformed associated Legendre functions ˜ P n m : ˜ P n | m| (sin " ) = (#1) m P n | m| (sin" ) (5) n = 0,1,2, 3, K m = 0, K, + n ˜ P 0 0 (sin " ) = 1 ˜ P 1 0 (sin " ) = sin" ˜ P 1 1 (sin" ) = cos" ˜ P 2 0 (sin" ) = 1 2 (3cos 2 " # 1) ˜ P 2 1 (sin" ) = 3cos" sin" ˜ P 2 2 (sin" ) = 3cos 2 " ˜ P 3 0 (sin" ) = 1 2 sin" (5sin 2 " # 3) ˜ P 3 1 (sin " ) = 3 2 cos" (5sin 2 " # 1) ˜ P 3 2 (sin" ) = 15cos 2 " sin" ˜ P 3 3 (sin" ) = 15cos 3 "

Transcript of Appendix A Recursions of associated Legendre functions€¦ · Appendix A Recursions of associated...

Page 1: Appendix A Recursions of associated Legendre functions€¦ · Appendix A Recursions of associated Legendre functions! P n m:! P 0 0(x)=1 (1) ! ! P n n(x)=(2n"1)1"x2P n"1 n"1(x) n"1

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"

Page 2: Appendix A Recursions of associated Legendre functions€¦ · Appendix A Recursions of associated Legendre functions! P n m:! P 0 0(x)=1 (1) ! ! P n n(x)=(2n"1)1"x2P n"1 n"1(x) n"1

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$

Page 3: Appendix A Recursions of associated Legendre functions€¦ · Appendix A Recursions of associated Legendre functions! P n m:! P 0 0(x)=1 (1) ! ! P n n(x)=(2n"1)1"x2P n"1 n"1(x) n"1

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#

Page 4: Appendix A Recursions of associated Legendre functions€¦ · Appendix A Recursions of associated Legendre functions! P n m:! P 0 0(x)=1 (1) ! ! P n n(x)=(2n"1)1"x2P n"1 n"1(x) n"1

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;