Mathematical Physics Vol 1

Chapter 5. Series Solutions of Differential Equations. Special functions

264

Definition Non-negative square root

vuu ut b Z a

p ( f n , f n )=

f 2 n ( x ) d x = √ I n

(5.221)

is called the norm of the function f n ( x ) and denoted by

vuu ut b Z a

∥ f n ∥ = p ( f n , f n )=

√ I n , i.e.

f 2 n ( x ) d x .

∥ f n ∥ =

(5.222)

Definition The function set f n (5.219), whose norm is equal to one, i.e. ∥ f n ∥ = vuu ut b Z a f 2 n ( x ) d x = 1

(5.223)

is called a normalized function set.

Definition The function set f n (5.219) that is both orthogonal and normalized, i.e.

b Z a

f m ( x ) f n ( x ) d x = δ mn

( f m , f n )=

(5.224)

is called an orthonormal function set on the interval x ∈ [ a , b ] .

In the previous relation δ i j represents Kronecker delta symbol. Some function sets, important for applications, are not orthogonal but have the following property b Z a p ( x ) f m ( x ) f n ( x ) d x = 0 , for m̸ = n . (5.225) In this case it is said that the function set f n (5.219) is orthogonal to the weight function p ( x ) , on the interval x ∈ [ a , b ] . The norm of functions from this set is defined by the following expression ∥ f n ∥ = vuu ut b Z a p ( x ) f 2 n ( x ) d x . (5.226)