The Chemistry Maths Book, Second Edition

15 Orthogonal expansions. Fourier analysis

15.1 Concepts

We saw in Section 7.6 that many functions can be expanded as power series and that,

indeed, some functions are defined by such series. In general, a functionf(x)of the

variable xcan be expanded in powers of xas a MacLaurin series if the function and its

derivatives exist atx 1 = 10 and throughout the interval 0 to x. The expansion is then

valid within the radius of convergence of the series.

The power series expansion of a function is a special case of a more general type of

expansion. Consider the function

(15.1)

We can regard each power of xas the function g

l

(x) 1 = 1 x

l

; that is, g

0

(x) 1 = 1 x

0

1 = 11 ,

g

1

(x) 1 = 1 x,g

2

(x) 1 = 1 x

2

and so on. The expansion is then

(15.2)

This formalism suggests that it may be possible to find other sets of functions{g

l

}

that can be used instead of simple powers, and that it may be possible in this way to

expand functions that cannot be expanded as power series.

The sets of functions that are of particular importance for expansions in series

consist of functions with the property of orthogonality. The theory of orthogonal

expansions is introduced in Section 15.2. Two examples of expansions in Legendre

polynomials, important in potential theory and in scattering theory, are discussed in

Section 15.3, with an application in electrostatics. Fourier series are developed in

Section 15.4, and used in Section 15.5 to solve the wave equation for the vibrating

string for a given set of initial conditions. Fourier transforms, essential for the analysis

of the results of diffraction experiments and in Fourier transform spectroscopy, are

discussed in Section 15.6.

15.2 Orthogonal expansions

We introduce the concept of expansions in sets of orthogonal functions by

demonstrating that the power series (15.1) can be written as a linear combination of

Legendre polynomials,

fx cPx cPx cPx cPx (15.3)

l

ll

()=+++=() () () ()

=

∑

00 11 22

0



∞

fx agx agx agx agx

l

ll

()=+++=() () () ()

=

∑

00 11 22

0



∞

fx a ax ax ax

l

l

l

()=+ + +=

=

∑

01 2

2

0



∞