1000 Solved Problems in Modern Physics

(Romina) #1

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

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