1000 Solved Problems in Modern Physics

6 1 Mathematical Physics

Recurrence formulas:

xPn′(x)−Pn′− 1 (x)=nPn(x)

Pn′+ 1 (x)−Pn′− 1 (x)=(2n+1)Pn(x) (1.32)

Orthonormal properties:

∫ 1

− 1

Pm(x)Pn(x)dx= 0 m

=n (1.33)

∫ 1

− 1

{Pn(x)}^2 dx=

2

2 n+ 1

(1.34)

Other properties:

Pn(1)= 1 ,Pn(−1)=(−1)n,Pn(−x)=(−1)nPn(x) (1.35)

Associated Legendre functions:

Differential equation:

(1−x^2 )y′′− 2 xy′+

{

l(l+1)−

m^2

1 −x^2

}

y= 0 (1.36)

Plm(x)=(1−x^2 )m/^2

dm

dxm

Pl(x) (1.37)

wherePl(x) are the Legendre polynomials stated previously,lbeing the positive
integer.

Plo(x)=Pl(x) (1.38)

andPlm(x)=0ifm>n (1.39)

Orthonormal properties:

∫ 1

− 1

Pnm(x)Plm(x)dx= 0 n

=l (1.40)

∫ 1

− 1

{Plm(x)}^2 dx=

2

2 l+ 1

(l+m)!

(l−m)!

(1.41)

Laguerre polynomials:

Differential equation:

xy′′+(1−x)y′+ny= 0 (1.42)

ifn= 0 , 1 , 2 ,...we get Laguerre polynomials given by