Legendre Polynomials, Associated Legendre Polynomials | PHYS 475, Study notes of Physics

Download Legendre Polynomials, Associated Legendre Polynomials | PHYS 475 and more Study notes Physics in PDF only on Docsity! ∫ − = l l n dx l xn xf l a π cos)( 1 ∫ − = l l n dx l xn xf l b π sin)( 1 ∫ − π − = l l l xin n dxe)x(f l c 2 1 mnnxdxmx δ π π π =∫ − coscos 1 mnnxdxmx δ π π π =∫ − sinsin 1 0sincos 1 =∫ − π π π nxdxmx xyyxz /tan;22 =θ+= niizLnz πθ 2ln ++= wzz ew ln= 2 cosh; 2 sinh zzzz ee z ee z −− + = − = )sin(cos yiyee xz += ∑ ∞ −∞= = n l xin necxf π )( ∑∑ ∞ = ∞ = ++= 11 02 1 sincos)( n n n n l xn b l xn aaxf ππ ( ) ( ) ( , ) . ( ) p q p q p q Γ Γ Β = Γ + 1 0 ( ) p tp t e dt ∞ − −Γ = ∫ ( 1) ! for integer 0, ( 1) ( ) for all real . p p p p p p p Γ + = > Γ + = Γ ∑ ∞ = −= 0 )( )( ! )( )( n n n ax n af xf ∫ −= x t dtexerf 0 22 )( π ( ) (1 ) . sin p p p π π Γ Γ − = nennn nn π2)1(! −=+Γ= ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 1/ 2 2 1 0 11 1 0 1 0 for 0, 0 : , 2 sin cos , , 1 , , . 1 p q p q p p q p q p q d p q x x dx y p q dx y π θ θ θ − − − − − ∞ + > > Β = Β = − Β = + ∫ ∫ ∫ Legendre polynomials Differential equation: ...2,1,0,0)1('2")1( 2 ==++−− lyllxyyx Normalization and orthogonality 1 1 2 . 2 1 l l llPP dx l δ′ ′ − = +∫ (1) 1 l P = ( )( ) 1 ( ) l l l P x P x= − − Rodriges formula ( ) ( )2 1 1 2 ! l l l l l d P x x l dx = − Generation function ( ) ( ) 1/ 2 2 0 , 1 2 .l l l x h xh h Ph ∞ − = Φ = − + =∑ Recurrence relations ( ) ( )1 22 1 1l l llP l xP l P− −= − − − 1 ,l l lxP P lP−′ ′− = 1 1,l l lP xP lP− −′ ′− = ( )2 11 ,l l lx P lP lxP−′− = − ( ) 1 12 1 .l l ll P P P+ −′ ′+ = − Associated Legendre polynomials Differential equation lm l y x m llxyyx ±±= = =       − −++−− ,...1,0 ...2,1,0 ,0 1 )1('2")1( 2 2 2 ( ) ( )0 ,l lP x P x= ( ) ( ) ( )1 . l m lm lm P x P x + − = − ( ) / 2 2 ( ) 1 ( ), m m lm lm d P x x P x dx = − ( ) ( ) 1 1 !2 ( ) ( ) . 2 1 ! lm l m ll l m dxP x P x l l m δ′ ′ − + = + −∫ y