Mathematical Methods for Physics and Engineering : A Comprehensive Guide

EIGENFUNCTION METHODS FOR DIFFERENTIAL EQUATIONS

Starting from the linearly independent functionsyn(x)=xn,n=0, 1 ,..., construct three

orthonormal functions over the range− 1 <x< 1 , assuming a weight function of unity.

The first unnormalised functionφ 0 is simply equal to the first of the original functions, i.e.

φ 0 =1.

The normalisation is carried out by dividing by

〈φ 0 |φ 0 〉^1 /^2 =

(∫ 1

− 1

1 × 1 du

) 1 / 2

=

√

2 ,

with the result that the first normalised functionφˆ 0 is given by

φˆ 0 =√φ^0

2

=

√

1

2.

The second unnormalised function is found by applying the above Gram–Schmidt orthog-
onalisation procedure, i.e.

φ 1 =y 1 −φˆ 0 〈φˆ 0 |y 1 〉.

It can easily be shown that〈φˆ 0 |y 1 〉=0,andsoφ 1 =x. Normalising then gives

φˆ 1 =φ 1

(∫ 1

− 1

u×udu

)− 1 / 2

=

√

3

2 x.

The third unnormalised function is similarly given by

φ 2 =y 2 −φˆ 1 〈φˆ 1 |y 2 〉−φˆ 0 〈φˆ 0 |y 2 〉

=x^2 − 0 −^13 ,

which, on normalising, gives

φˆ 2 =φ 2

(∫ 1

− 1

(

u^2 −^13

) 2

du

)− 1 / 2

=^12

√

5

2 (3x

(^2) −1).
By comparing the functionsφˆ 0 ,φˆ 1 andφˆ 2 with the list in subsection 18.1.1, we see that
this procedure has generated (multiples of) the first three Legendre polynomials.
If a function is expressed in terms of anorthonormalbasisφˆn(x)as
f(x)=
∑∞
n=0
cnφˆn(x) (17.10)
then the coefficientscnare given by
cn=〈φˆn|f〉=
∫b
a
φˆ∗n(x)f(x)ρ(x)dx. (17.11)
Note that this is true only if the basis is orthonormal.