Mathematical Physics Vol 1

Chapter 5. Series Solutions of Differential Equations. Special functions

274

The integral in this equation can be partially integrated several times 19 , which yields + 1 Z − 1 ( 1 − x ) n ( 1 + x ) n d x = =( 1 − x ) n ( 1 + x ) n + 1 n + 1 + 1 − 1 + n n + 1 + 1 Z − 1 ( 1 − x ) n − 1 ( 1 + x ) n + 1 d x =

+ 1 Z

( 1 + x ) n + 2 n + 2

n n + 1

( 1 − x ) n − 2 ( n − 1 )

d x =

=

− 1

+ 1 Z

n n + 1

n − 1 n + 2

− 1 ( 1 − x ) n − 2 ( 1 + x ) n + 2 d x = ··· =

=

+ 1 Z

+ 1 Z

n n + 1

n − 1 n + 2 ···

1 2 n

n ( n − 1 ) ··· 1 ( n + 1 )( n + 2 ) ··· ( 2 n )

2 n d x

2 n d x

( 1 + x )

( 1 + x )

=

=

.

− 1

− 1

Given that

+ 1

+ 1 Z

( 1 + x ) 2 n + 1 2 n + 1

2 2 n + 1 2 n + 1

2 n d x

( 1 + x )

=

=

,

− 1

− 1

for the square of the norm we finally obtain

2 · 2 2 n 2 n + 1

1 ( 2 n n ! )

n !1 · 2 ··· n 1 · 2 ··· n · ( n + 1 ) ··· ( 2 n )

2 2 n + 1

2

∥ P n ( x ) ∥

2 ( 2 n ) !

=

=

.

We have thus proved that Legendre polynomials are orthogonal, but that they are not normalized.

Problem 189 Prove that for each fixed n = 0 , 1 ,... , Bessel functions J n ( λ 1 n x ) , J n ( λ 2 n x ) , . . . , form a set of orthogonal functions on the interval 0 ≤ x ≤ R , with respect to the weight function p ( x )= x , i.e. that R Z 0 xJ n ( λ kn x ) J n ( λ mn x ) d x =   0 , m̸ = k , R 2 2 J 2 n + 1 ( λ mn R ) , m = k . (5.243)

Problem 190

Prove that

+ ∞ Z

=

0 , n̸ = m , n ! √ 2 π , n = m ,

x 2 / 2 d x

He m ( x ) He n ( x ) e −

(5.244)

− ∞

19 Given that R u , d v = uv − R v , d u , we assumed that in this case u =( 1 − x ) 2 , and d v =( 1 + x ) n d x .