Colatitude dependence Solution for colatitude function generates Legendre polynomials and associated functions The polynomials occur when m=0 in n dependence. t=cos(0) Pn()= 2 nl dt 02/1302 12.540Lec03
02/13/02 12.540 Lec 03 6 Colatitude dependence • Solution for colatitude function generates Legendre polynomials and associated functions. • The polynomials occur when m=0 in λ dependence. t=cos( θ ) Pn ( t ) = 1 2 n n! d n d t n ( t 2 −1) n
Legendre Functions Low order functions P()=t Arbitrary n P2()=(312-1) 2 values are P3()=(5t3-3) generated by recursive P24(t)=(35t-3012+3 algorithms 02/1302 12.540Lec03
02/13/02 12.540 Lec 03 7 Legendre Functions • Low order functions. Arbitrary n values are generated by recursive algorithms Po ( t ) = 1 P1( t ) = t P2 ( t ) = 1 2 (3 t 2 −1) P3 ( t ) = 1 2 (5 t 3 − 3 t ) P4 ( t ) = 1 8 (3 5 t 4 − 30 t 2 +3)
Associated Legendre Functions The associated functions satisfy the following equation Pn(t)=(-1y(1-2)m2 d 2P() The formula for the polynomials Rodriques formula, can be substituted 02/1302 12.540Lec03
02/13/02 12.540 Lec 03 8 Associated Legendre Functions • The associated functions satisfy the following equation • The formula for the polynomials, Rodriques’ formula, can be substituted Pnm ( t ) = ( −1) m (1 − t 2 ) m/2 d m d t m Pn ( t )