Page 219 - DMTH504_DIFFERENTIAL_AND_INTEGRAL_EQUATION

Differential and Integral Equation

Notes n 1
x
( 1) n H m ( ) d n 1 e x 2 dx
dx

d n 1 x 2
x
= ( 1) 2mH m 1 ( ) n 1 e dx
dx
[since e x 2 and all its derivatives
vanish for infinite x and H = 2n H ]
n n 1
d n 1 2
= ( 1) n 1 2m H m 1 ( ) 1 e x dx n m
x
dx n
proceeding similarly again and again
d n m x 2
n m m
= ( 1) 2 m ! H 0 ( ) e dx n m
x
dx n m
d n m x 2
m
n m
x
= ( 1) 2 m ! n m e dx [  H ( ) 1]
dx 0
d n m 1 2
m
= ( 1) n m 2 m ! e x
dx n m 1
= 0

x
Now H 2 ( )e x 2 dx = H ( ) d n e x 2 dx integrating as above n times
x
n n n
dx
2
n
x
x
= 2 n H 0 ( )e dx
n
= 2 n ! e x 2 dx

2
x
n
= 2 n !2 e dx
0
n
= 2 n ! .
The functions of the orthogonal system are
H n ( )e x 2 2
x
(x) = , (n 0,1,2,...)
n n
2 n !
(d) Orthogonality of Laguerre Polynomials

The Laguerre Polynomials L (x) given by
n
x d n n x
L (x) = e (x e )
n n
dx

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