The Chemistry Maths Book, Second Edition

420 Chapter 15Orthogonal expansions. Fourier analysis

The norm of a function is sometimes interpreted as the ‘magnitude’ or ‘length’ of the

function. The coefficients in the expansion (15.11) off(x)are then given by

(15.14)

It is often more convenient if the functions of the expansion set are normalized, as

well as orthogonal. Normalization is achieved by dividing each function by its norm,

(15.15)

The resulting set of functions is an orthonormalset,

(15.16)

The expansion (15.11) then becomes

(15.17)

and (15.14) for the expansion coefficients becomes

(15.18)

The concept of orthogonal expansions is readily generalized for functions of more

than one variable.

Completeness of orthogonal sets

A number of orthogonal sets have been discussed in previous chapters; in particular,

the several ‘special functions’ in Chapter 13 and the solutions of the Schrödinger

equation (eigenfunctions of the Hamiltonian) in Chapters 12 and 14. These sets share

the property of completeness:

An orthonormal set{g(x)}defined in the intervala 1 ≤ 1 x 1 ≤ 1 bwith respect to weight

functionw(x)is called complete if ‘every’ functionf(x)defined in the interval

can be represented by a linear combination of the functions of the set,

(15.19)

This expression of completeness is sufficient for most purposes. We consider two

qualifications.

fx cg x c g xfx xdx

n

nn n

a

b

n

() ()

*

=,=() ()()

=

∑

0

∞

Z w

cgxfxxdx

n

a

b

n

=Z *() () ()w

fx c

g

x

n

n n

()= ()

=

∑

0

∞

Z

a

b

nn mn

gxgx xdx*() () ()w =

,

δ

n

n

n

g x

g

()= gx()

1

c

g

gxfx xdx

n

n

a

b

n

=

1

2

Z

*

()() ()w