10 Bessel differential equation

TABLE OF CONTENTS

1. Introduction

2. Bessel functions

3. The recurrence relations

4. Bessel Functions of integer order – Integral representation

4.1 The generating function

5. Bessel function of the second kind

5.1 Hankel functions

6. Orthogonality of Bessel functions

6.1 Expansion of a function in terms of Bessel functions

7. Asymptotic behaviour of Bessel Functions

LEARNING OBJECTIVES

1. Importance of Bessel differential equation is stressed and its derivation from Laplace equation sketched.
2. Bessel functions are introduced as solutions of Bessel equation.
3. Recurrence relations for the Bessel functions are obtained.
4. Discussion is specialized to Bessel functions of integer order; integral representation is obtained and the generating function derived.
5. Bessel function of the second kind are introduced and discussed in detail; the related Hankel functions are defined.
6. Idea of Orthogonality of Bessel functions is introduced and expansion of a function in terms of Bessel functions obtained.
7. The asymptotic behaviour of Bessel functions is briefly discussed.

B e s s e l  d i f f e r e n t i a l  e q u a t i o n

1. Introduction

Apart from the Legendre differential equation the other equally important and ubiquitous equation in physics and related sciences is the differential equation due to Bessel. It appears in the study of problems involving circular membranes and circular disks etc. Like the Legendre equation, Bessel equation also arises in the solution of the Laplace equation by the method of separation of variables, in this case in cylindrical coordinates. In cylindrical coordinates Laplace equation takes the form

This is the Bessel differential equation.

The equation is defined for all (complex) values of the parameter ν. The most important cases are for ν equal to an integer or half integer. In most problems V is required to be a single valued function which implies that ν must be an integer. In this case Bessel’s functions are known as cylindrical functions or cylindrical harmonics. Bessel functions with half-integer order are obtained in the solution of Helmholtz equation in spherical coordinates.

2. Bessel functions

Let us make a change of variable

We attempt a series solution of the form

On substituting in equation (3) we get

Now compare the coefficients of various powers of y and we get

result for this case is obtained by simply replacing ν by –ν. We thus obtain two series solutions of the Bessel equation, equation (8) and

These two solutions may or may not be linearly independent.

We now define the Bessel function of the first kind of order ν to be the function defined by the series

is thus an entire or integral function.

Since equations (8) and (9) represent solutions of Bessel equation, so does

This is a solution with two independent constants. However, it will represent the complete solution if the two solutions are linearly independent. For ν = 0 the two solutions are identical; so certainly not linearly independent. The same is true if ν is an integer. For if ν = n, an integer, we have

3. The recurrence relations

In the first sum write the i = 0 term separately and in the second sum replace i by i – 1:

On using the properties of the gamma function

We thus have the first recurrence relation

Exactly in a similar fashion we have

Few more recurrence relations can be obtained by combining these two.

4. Bessel Functions of integer order – Integral representation

Many of the formulae take a simpler form for Bessel functions of integer order. There are also some relations which are specific to Bessel functions of integer order only. Like the case of Legendre polynomials there is the Schlӓfli contour integral for the Bessel functions as well, which we state below without proof:

4.1 The generating function

5. Bessel function of the second kind

On applying the L’Hospital’s rule for finding the limit, we have

Let us now find the series expansion for this function. Substitute the series for the Bessel function into the above equation (16):

Then

The other terms of the sum can be treated in a straightforward manner.  The result is

Collecting all the terms together we have, for positive integer n,

5.1 Hankel functions

From the linear combination of the Bessel functions of the first and second kind we define another equivalent set of functions, the Bessel functions of the third kind, more often called the Hankel functions of order n through

    Here i stands for √−1. Since Bessel functions of first and second kind are linearly independent, so are the Hankel functions; hence they provide an alternative pair of solutions to the Bessel equation. The usefulness of the various sets depends on the asymptotic behaviour of these functions near the origin and infinity.

6. Orthogonality of Bessel functions

Bessel functions provide an orthonormal set in a certain sense which we will now explain. Bessel functions in a way are like sine and cosine functions. They are oscillating functions having infinite number of zeros. But, unlike trigonometric functions they are oscillating with decreasing amplitude. Also the zeros of Bessel functions are not equally spaced.

Or, equivalently,

Apply this result to both the terms of the above equation and we obtain

Or

On integrating over z from 0 to a, we obtain

Next use the recurrence relation

Hence

Thus we have proved the theorem

6.1 Expansion of a function in terms of Bessel functions

Now we use the orthonormality relation (22) on the right hand side, so that

the required result.

7. Asymptotic behaviour of Bessel Functions

In view of the importance of the behaviour of the Bessel functions for large and small values of the arguments, we quote below the results without giving any proof.

For large values of the argument z (  z → ∞), we have

For z→ 0, the behaviour of the Bessel functions can be read from the power series expansion.  We have

SUMMARY

• We stress the importance of Bessel differential equation and Bessel functions in physics and sketch the derivation of the equation from Laplace equation.
• We next obtain the series solution of the Bessel equation which defines Bessel functions of first kind.
• We derive the recurrence relations for the Bessel functions.
• Next we specialize to Bessel functions of integer order; obtain the integral representation and then derive the generating function.
• We introduce and discuss in detail Bessel function of the second kind which together with Bessel function of the first kind provide a complete solution of the Bessel equation and also define the related Hankel functions.
• We explain in what sense the Bessel functions are orthogonal and prove the orthogonality and obtain expansion of a function in terms of Bessel functions.
• The asymptotic behaviour of various Bessel functions is important in many applications; we simply describe the asymptotic behaviour of these functions without derivation.