Mathematical Physics Vol 1

5.10 Examples

275

i.e. Hermite functions form a set of orthogonal functions, with respect to the weight function p ( x )= e − x 2 / 2 .

Problem 191 Prove that Laguerre functions satisfy the following equations a) ∞ Z 0 L α m ( x ) L α n ( x ) e − x d x = δ mn ,

(5.245)

b)

=  

∞ Z 0

0 ,

n̸ = m ,

L α

α n ( x ) e −

x x α d x

( n + α ) ! n !

m ( x ) L

(5.246)

, n = m .

Legendre polynomials

Problem 192 Prove that the function G ( x , t )=( 1 − 2 xt + t 2 ) − 1

2 generates Legendre polynomials,

i.e.

∞ ∑ n = 0

G ( x , t )= 1 − 2 xt + t 2 − 1

n .

P n ( x ) t

2 =

Solution The observed function has two singularities G ( x , t ) , namely the two zeroes of the polynomial 1 − 2 xt + t 2 : t 1 , 2 = x ± i p 1 − x 2 . As the absolute value of both solutions | t 1 | = | t 2 | = 1, we can conclude that the function can be expanded into Taylor series in the vicinity of point t = 0, as the series is convergent for | t | < 1. If | 2 xt − t 2 | < 1 we have the following expansion ( 1 − 2 xt + t 2 ) − 1 2 = 1 − ( 2 xt − t 2 ) − 1 2

( − 1 ) k − ( 2 k − 1 ) !! ( 2 k ) !!

1 / 2 k

∞ ∑ k = 0 ∞ ∑ k = 0 ∞ ∑ k = 0

( 2 xt − t 2 ) k

=

(5.247)

( 2 xt − t 2 ) k

=

( 2 k ) ! 2 2 k k ! k !

( 2 xt − t 2 ) k .

=

If the condition

| t | ( 2 | x | + | t | ) < 1 ,

(5.248)