18

Special functions

In the previous two chapters, we introduced the most important second-order

linear ODEs in physics and engineering, listing their regular and irregular sin-

gular points in table 16.1 and their Sturm–Liouville forms in table 17.1. These

equations occur with such frequency that solutions to them, which obey particu-

lar commonly occurring boundary conditions, have been extensively studied and

given special names. In this chapter, we discuss these so-called ‘special functions’

and their properties. In addition, we also discuss some special functions that are

not derived from solutions of important second-order ODEs, namely the gamma

function and related functions. These convenient functions appear in a number

of contexts, and so in section 18.12 we gather together some of their properties,

with a minimum of formal proofs.

18.1 Legendre functions

Legendre’s differential equation has the form

(1−x^2 )y′′− 2 xy′+(+1)y=0, (18.1)

and has three regular singular points, atx=− 1 , 1 ,∞. It occurs in numerous

physical applications and particularly in problems with axial symmetry that

involve the∇^2 operator, when they are expressed in spherical polar coordinates.

In normal usage the variablexin Legendre’s equation is the cosine of the polar

angle in spherical polars, and thus− 1 ≤x≤1. The parameteris a given real

number, and any solution of (18.1) is called aLegendre function.

In subsection 16.1.1, we showed thatx= 0 is an ordinary point of (18.1), and so

we expect to find two linearly independent solutions of the formy=

∑∞

n=0anx

n.

Substituting, we find

∑∞

n=0

[

n(n−1)anxn−^2 −n(n−1)anxn− 2 nanxn+(+1)anxn

]

=0,