9 Legendre differential equation and polynomials

TABLE OF CONTENTS

1. Introduction

2. Solution of the equation

2.1 An alternative method

3. The Legendre polynomials

3.1 Rodrigue’s formula

3.2 Zeros of Legendre polynomials

3.3 Generating function for Legendre polynomials

3.4 The recurrence relations

4. Integrals involving products of Legendre polynomials

4.1 Orthonormality of Legendre polynomials

5. The complete solution

LEARNING OBJECTIVES

1. Importance of Legendre equation in physics is explained and its derivation from Laplace equation is sketched.

2. Series solutions of the Legendre equation are obtained for the case of integer values of the parameter n.

3. It is explained as to how one solution becomes a polynomial of order n, called the Legendre polynomials.

4. Detailed study of the properties of the Legendre polynomials is undertaken. Rodrigue’s formula is derived. Properties of zeroes of Legendre polynomials are studied. The generating function is derived and the recurrence relation obtained.

5. Integrals involving products of Legendre polynomials are obtained from which follow the orthonormality property of these polynomials.

6. Finally the complete solution of the Legendre equation is discussed briefly.

L e g e n d r e  d i f f e r e n t i a l  e q u a t i o n  a n d  P o l y n o m i a l s

1. Introduction

Legendre differential equation is one of the most important and ubiquitous equations in physics. One of the ways it appears in physics is via the solution of the Laplace equation

The solutions of the Laplace equation are the harmonic functions which are of the greatest importance in every branch of physics, particularly electromagnetism, gravitation, fluid dynamics and heat conduction. When written in spherical polar coordinates, the equation becomes (we will learn the details in the topic on partial differential equations)

This is the associated Legendre equation. In the case of greatest interest m can take only integer values 0, 1, 2, ….n. Finally if we consider the special case of m = 0, we obtain the Legendre equation.

2. Solution of the equation

Although the parameter n can take any value, even complex values, the case of general interest is one in which n is a non-negative integer, and that is the case we will study. We can easily verify that the singularity at = ±1 is regular with the exponent taking value 1, and the one at infinity is also regular with the exponent taking values, –n and n + 1. The origin is an ordinary point of the equation. If we rewrite the equation in the form

we see that

The radius of convergence of both the series is R = 1.  We seek a solution of the form

The radius of convergence of this series will be at least equal to 1. On differentiating this series twice we obtain

Continuing in this way, we obtain for the even and odd indices respectively

The general solution (6) of the Legendre equation (2) can therefore be written as

2.1 An alternative method

We could have started with the point at infinity which is a regular singular point with exponents –n and n + 1. In that case we attempt a series solution of the form

The solution with the exponent –n is

Thus we have again found two linearly independent solutions of the Legendre equation. The first solution is a polynomial solution and is valid for all z while the second is an infinite power series convergent for |z| > 1.

3. The Legendre polynomials

Let us take the constant a in equation (15) to be

3.1 Rodrigue’s formula

3.2 Zeros of Legendre polynomials

But the solution of the initial value problem is unique and ≡ 0 is a solution. Hence no solution, in particular the Legendre polynomials, can have a double root. Therefore all the n roots of the Legendre polynomial of order n are real, distinct and lie in the open interval (-1, 1).

3.3 Generating function for Legendre polynomials

If we use Cauchy’s formula for the nth derivative of an analytic function, discussed in the study of complex variables, for the Rodrigue’s formula, we obtain

Here C is any closed contour surrounding the point t = z. This is called Schläfli integral formula. For the contour C, let us take the circle

Then on the contour C

This is known as the Laplace first integral for Legendre polynomials.

3.4 The recurrence relations

Using the generating function we can deduce certain recurrence relations between Legendre polynomials and their derivatives of different orders. We can easily verify that the generating function satisfies the differential equation

Now using equation (23) we get

Similarly on using the relation

in equation (23) we get the second recurrence relation

If we now differentiate equation (24) with respect to z we obtain

We can use these three recurrence relations to derive a few more such relations. For example, if we multiply equation (25) by z and subtract equation (26) from it, we get

4. Integrals involving products of Legendre polynomials

Now by using recurrence relation (27) we get

This is the required result for m≠n  . When m=n we make use of recurrence relations (24) and (25) from which we get

We now integrate to obtain

This is the result for the integral of the square of Legendre polynomials.

4.1 Orthonormality of Legendre polynomials

The two relations can be written together by introducing the Kronecker delta function:

Thus the Legendre polynomials form an orthonormal set over the interval −1 ≤ ≤ 1. This was the reason in the first place for choosing the value for the constant a in the solution of the Legendre equation according to equation (17). As we saw in the introduction, Legendre equation arises in the solution of Laplace equation where the independent variable z actually stands for cos(θ). As θ varies from 0 to 2π, cos(θ) varies from (-1, 1). So the range (-1, 1) is a “natural” range for the variable in Legendre polynomials.

5. The complete solution

The Legendre equation (2) has regular singularities at = ±1 where the exponent is 1. Hence the exponent difference is zero at both the singularities; so one of the solutions has a logarithmic singularity. To find this solution we use the usual method of obtaining the second solution when one solution is known. For this purpose we make the substitution

Substitute this in equation (2) and we find that v(z) satisfies the equation

SUMMARY

• We explain the importance of Legendre equation in physics and sketch its derivation from Laplace equation.
• We obtain the series solutions of the Legendre equation for the case of integer values of the parameter n appearing in the equation.
• We explain as to how one solution becomes a polynomial of order n, which is called the Legendre polynomial.
• We undertake a detailed study of the properties of the Legendre polynomials; derive Rodrigue’s formula, study the properties of zeroes of Legendre polynomials, derive the generating function and obtain the recurrence relations.
• Next we obtain integrals involving products of Legendre polynomials and thereby obtain the orthonormality property of these polynomials.
• Finally we briefly discuss the complete solution of the Legendre equation.