leading to

√

β2 ±

sech ±(t ’ β1 z)ei±|β2 |z/2 eik0 z’iω0 t .

E(z, t) = (7.128)

γ

’1

This equation describes a pulse propagating at β1 , which is the group ve-

locity.

Chapter 8

Special Functions I

In solving Laplace™s equation by the method of separation of variables we

come across the most important of the special functions of mathematical

physics. These functions have been studied for many years, and books such as

the Bateman manuscript project1 summarize the results. Any serious student

theoretical physics needs to be familiar with this material, and should at least

read the standard text: A Course of Modern Analysis by E. T. Whittaker

and G. N. Watson (Cambridge University Press). Although it was originally

published in 1902, nothing has superseded this book in its accessibility and

usefulness.

In this chapter we will focus only on the properties that all physics stu-

dents should know by heart.

8.1 Curvilinear Co-ordinates

Laplace™s equation can be separated in a number of coordinate systems.

These are all orthogonal systems in that the local coordinate axes cross at

right angles.

1

The Bateman manuscript project contains the formul¦ collected by Harry Bateman,

who was professor of Mathematics, Theoretical Physics, and Aeronautics at the California

Institute of Technology. After his death in 1946, several dozen shoe boxes full of ¬le cards

were found in his garage. These proved to be the index to a mountain of paper contain-

ing his detailed notes. A subset of the material was eventually published as the three

volume series Higher Transcendental Functions, and the two volume Tables of Integral

Transformations, A. Erdelyi et al. eds.

215

216 CHAPTER 8. SPECIAL FUNCTIONS I

To any system of orthogonal curvilinear coordinates is associated a metric

of the form

ds2 = h2 (dx1 )2 + h2 (dx2 )2 + h2 (dx3 )2 . (8.1)

1 2 3

√

This expression tells us the distance ds2 between the adjacent points

(x1 + dx1 , x2 + dx2 , x3 + dx3 ) and (x1 , x2 , x3 ). In general, the hi will depend

on the co-ordinates xi .

The most commonly used orthogonal curvilinear co-ordinate systems are

plane polars, spherical polars, and cylindrical polars. The Laplacian also

separates in plane elliptic, or three-dimensional ellipsoidal coordinates and

their degenerate limits, such as parabolic cylindrical co-ordinates ” but these

are not so often encountered, and we refer the reader to more comprehensive

treatises, such Morse and Feshbach™s Methods of Theoretical Physics.

Plane Polar Co-ordinates

y

P

r

θ x

Plane polar co-ordinates.

Plane polar co-ordinates have metric

ds2 = dr 2 + r 2 dθ2 , (8.2)

so hr = 1, hθ = r.

8.1. CURVILINEAR CO-ORDINATES 217

Spherical Polar Co-ordinates

z

P

r y

θ

φ x

Spherical co-ordinates.

This system has metric

ds2 = dr 2 + r 2 dθ2 + r 2 sin2 θdφ2 , (8.3)

so hr = 1, hθ = r, hφ = r sin θ,

Cylindrical Polar Co-ordinates

z

r P

y

z

θ x

Cylindrical co-ordinates.

These have metric

ds2 = dr 2 + r 2 dθ2 + dz 2 , (8.4)

so hr = 1, hθ = r, hz = 1.

218 CHAPTER 8. SPECIAL FUNCTIONS I

8.1.1 Div, Grad and Curl in Curvilinear Co-ordinates

It is very useful to know how to write the curvilinear co-ordinate expressions

for the common operations of the vector calculus. Knowing these, we can

then write down the expression for the Laplace operator.

The gradient operator

We begin with the gradient operator. This is a vector quantity, and to

express it we need to understand how to associate a set of basis vectors with

our co-ordinate system. The simplest thing to do is to take unit vectors ei

tangential to the local co-ordinate axes. Because the coordinate system is

orthogonal, these unit vectors will then constitute an orthonormal system.

eθ

er

Unit basis vectors in plane polar co-rdinates.

The vector corresponding to an in¬nitsimal co-ordinate displacement dxi is

then given by

dr = h1 dx1 e1 + h2 dx2 e2 + h3 dx3 e3 . (8.5)

Using the orthonormality of the basis vectors, we ¬nd that

ds2 ≡ |dr|2 = h2 (dx1 )2 + h2 (dx2 )2 + h2 (dx3 )2 , (8.6)