The Chemistry Maths Book, Second Edition

15.3 Two expansions in Legendre polynomials 421

(1) By everyfunction is meant not only continuous functions but the more general

class of piecewise continuousfunctions that have a finite number of finite

discontinuities and a finite number of maxima and minima in the interval.

1

(2) By the representationof the function is meant that the function can be approximated

arbitrarily closely by a series

f(x) 1 ≈ 1 f

k

(x) 1 = 1 c

0

g

0

(x) 1 + 1 c

1

g

1

(x) 1 +1-1+ 1 c

k

g

k

(x) (15.20)

and the limit of the series is such that

(15.21)

The series is said to converge tof(x)in the mean, and the function can in practice be

replaced by the series even though (15.21) means that the series need not converge

tof(x)for every x. It is this convergence in the mean that allows the representation

of some discontinuous functions, but care must sometimes be taken when equating

derivatives and integrals of the series to the derivatives and integrals of the function it

represents.

15.3 Two expansions in Legendre polynomials

We saw in the previous section that a power series in xcan be represented as a

linear combination of Legendre polynomials. Two examples in the physical sciences

are

(15.22)

of importance in potential theory (electrostatics and gravitational theory), and

(15.23)

of importance in scattering theory, where thej

l

(t)are spherical Bessel functions

(equation (13.59)). The series (15.22) converges when |x| 1 < 11 and |t| 1 < 11 , and is

derived in Example 15.2. The series (15.23) converges when|x| 1 < 11 for all values of t,

and its derivation is discussed in Example 15.3.

elijtPx

itx

l

l

ll

=+

=

∑

0

21

∞

()()()

() ()121 1

212

0

−+ = , −≤≤+

−/

=

∑

xt t t P x x

l

l

l

∞

Z

a

b

k

fx f x xdx k

2

() () ()− 0









w →→as ∞

1

This was first proved by Dirichlet for the Fourier series (‘Dirichlet conditions’). Gustav Peter Lejeune-

Dirichlet (1805–1859), German mathematician, was influenced by the work of Fourier whilst a student in Paris in

the early 1820’s.