THE  LEGENDRE  POLYNOMIALS

 

The Generating Function

 

The origin of the Legendre polynomials lies in the treatment of potential problems.  For example, the potential at the point r due to a point charge  located at the point  is given by

 

 

                                                      (1.1)

 

Assuming that , we write

 

                                                               (1.2)

 

or, introducing the notation  and ,

 

                                                                                       (1.3)

 

It is often desirable to expand this potential in powers of h for a given value of x.  So we write

 

                                                              (1.4)

 

The expansion coefficients  are a set of polynomials of order n, known as the Legendre polynomials, and the function  is known as the generating function for the Legendre polynomials.

 

 

Low-order Legendre Polynomials

 

            The Legendre polynomials of low order may be readily obtained by writing down the first few terms of the binomial expansion of the generating function:

 

       (1.5)

 

                                         (1.6)

whence, by comparing coefficients of h, we obtain

 

                                                            (1.7)

 

Incidentally, reverting to the notation  and , we see that the potential due to a point charge  may be written as

 

                                                (1.8)

 

or                                                (1.9)

 

The term in  is known as the monopole term, and represents the potential due to a point charge  placed at the origin (note that  is actually located at the point  ).  The term involving the quantity  is known as the dipole term,  being the dipole moment of the point charge .  The next term is known as the quadrupole term, and so on.  This multipole expansion of the potential is not often employed in the case of a single point charge, but provides a useful approximation in the case of an extended charge distribution.  The first term in the expansion is obtained by replacing the whole charge by a point charge of equal magnitude at the origin.  If the charge distribution is spherically symmetric, and the center of symmetry is chosen as the origin, this first term actually gives the potential exactly, despite the apparent crudity of the procedure.  In general, however, the monopole term alone gives a rather poor approximation of the potential.  For example, if the total charge is equal to zero, the potential does not necessarily vanish, as would be suggested by truncating the series after the monopole term.  The dipole term, at the very least, must also be included.  However, for some charge distributions, this term is also zero, and we must resort to the quadrupole term to obtain a meaningful approximation of the potential.

 

Special Properties of the Legendre Polynomials

 

            If we put  in Equation (1.4), we obtain

 

                                                                                          (2.1)

 

                                                                                                         (2.2)

 

 Note, further, that

                                               (2.3)

Comparing coefficients of , we see that

                                                                                                         (2.4)

 

or                                                                                            (2.5)

 

This result is known as the Parity property.  It can be used to obtain an important property of the Legendre polynomials, which can be seen as follows:

 

Let                                                                                                            (2.6)

 

By virtue of the parity property, we then have

 

                                                                                            (2.7)

 

whence, on comparing coefficients of , we have

 

                                                                                           (2.8)

 

This means that if n is even, only even values of r are permitted, i.e. the Legendre polynomial  contains only even powers of x.  Similarly, if n is odd,  contains only odd powers of x

 

Series Form of the Legendre Polynomials

 

            We have already obtained analytic expressions for some of the lower-order Legendre polynomials.  In the general case, we use the binomial expansion of the generating function, as follows:

 

                        (2.9)

 

                                            (2.10)

 

                                                                 (2.11)

 

Using the binomial expansion once again, we get

 

                                    (2.12)

 

                                               (2.13)

 

For purposes of comparison, the Right Hand Side of Equation (2.13) must be modified in such a way that the exponent of h becomes equal to n.  To do this, we change n to , which gives

 

                                 (2.14)

 

where  is equal to  if n is even and  if n is odd.  The upper limit on r is dictated by the presence of the term  in the denominator of Equation (2.14), which requires that .  Comparing coefficients of  in Equation (2.14), we obtain the series representation of the Legendre polynomials:

 

                                                     (2.15)

 

 

Rodrigues’ Formula

 

            The Legendre polynomials can be cast into a more useful form by writing Equation (2.15) as follows:

 

                                            (2.16)

 

and using the result

                                                                                 (2.17)

 

 

It follows from Equations (2.16) and (2.17) that

 

                                                                  (2.18)

 

                                                                  (2.19)

 

where we have extended the upper limit in the summation from  to n.  The justification for this procedure is that if r  >  , then the exponent .  The result

 

                                                                                                      (2.20)

 

then shows that each of the additional terms is zero.  Now the summation in Equation (2.19) is immediately recognized as the binomial expansion of , so we obtain

 

                                                                                (2.21)

 

This result is known as Rodrigues’ formula for the Legendre polynomials.  It shows explicitly that  is a polynomial of order n.  For the highest power of x in the expansion of  is , and in view of Equation (2.17) it follows that the highest power of x in the expression for  is , i.e.  is a polynomial of order n.

Evaluation of Integrals Involving Legendre Polynomials

 

            For an arbitrary function  defined on the interval , consider the integral

 

                                                                                                          (3.1)

 

Using the Rodrigues’ formula, this may be written as

 

                               (3.2)

 

Integrating by parts, we obtain

 

                               (3.3)

 

The first term vanishes at both limits, so we obtain

 

                                                                                  (3.4)

 

and it is clear that n partial integrations yields the final result

 

                                                (3.5)

 

Orthogonality of the Legendre Polynomials

 

            The result we have just derived can be used to obtain a very important property of the Legendre polynomials.  Let us suppose that .  Equation (3.5) then becomes

 

                    (4.1)

 

If m and n are different, one of them must be less than the other, and we lose nothing in generality by assuming that m < n.  The highest power in the expansion of  is .  The result

 

                                                                                           (4.2)

 

then shows that

 

                                                                              (4.3)

 

Equation (4.3) is known as the orthogonality relation, and the Legendre polynomials  are said to form an orthogonal sequence on the interval .

 

The case  can be investigated by inserting the result

 

                                                                                                       (4.4)

 

into Equation (4.1), to give

 

                                                                              (4.5)

 

The integral on the Right Hand Side of Equation (4.5) may be evaluated by recourse to the Gamma Function

 

                                                                                                            (4.6)

the Beta Function

 

                                                                                              (4.7)

the relations

 

                                                                                (4.8)

 

                                                                                                        (4.9)

 

and the duplication formula

 

                                                                                         (4.10)

 

Making the substitution , we may write

 

                                                   (4.11)

 

                                                    (4.12)

 

Replacing n by  in the duplication formula (4.10), we get

 

                                                                              (4.13)

 

                                                  (4.14)

 

Substituting Equation (4.14) and Equation (4.12) into Equation (4.5), we get

 

                                                                    (4.15)

 

which may be combined with Equation (4.3) in the form

 

                                                                              (4.16)

 

Alternatively, the orthogonality condition above may be obtained as follows:

 

                                                    (4.17)

 

or                                                    (4.18)

 

The integral on the Right may be evaluated by making the substitution

 

                                                                               (4.19)

 

Putting  in Equation (4.18), we get

 

                                            (4.20)

 

Since we have made the assumption that , we may expand the logarithm to give

 

               (4.21)

 

                                                                      (4.22)

 

and, comparing coefficients of , we obtain the result

 

                                                                                      (4.23)

 

Series of Legendre Polynomials

 

            It is sometimes desirable to express a given function , defined on the interval  in the form of a series of Legendre polynomials:

 

                                                                                                      (5.1)

 

A set of functions for which such an expansion can be made is known as a complete set.  The expansion coefficients  can be found by multiplying both sides of Equation (5.1) by , integrating with respect to x from  and using the orthogonality relation.  This gives

 

                                                                         (5.2)

 

                                                       (5.3)

 

                         (5.4)

 

It is assumed, of course, that the function  is sufficiently well behaved that the integral on the Right Hand Side of Equation (5.4) can be evaluated.

 

 

Recurrence Relations for the Legendre Polynomials

 

            A recurrence relation is a relation between functions and/or their derivatives of different order.  To obtain such relations for the Legendre polynomials, we start with the generating function

                                                                        (6.1)

 

and differentiate with respect to h:

 

                                                                         (6.2)

 

Multiplying both sides by  and using Equation (6.1), we get

 

                                                          (6.3)

 

and comparing coefficients of , we obtain the three-term recurrence relation

 

                                            (6.4)

 

or                                                     (6.5)

 

 

            Since the low-order Legendre polynomials are already known [see Equation (1.7)], this recurrence relation allows us to obtain expressions for Legendre polynomials of any order.  For example, if we put  in Equation (6.5), we get

 

                                                          (6.6)

 

                                                                                                              (6.7)

 

Higher order Legendre polynomials may be obtained by putting , etc. in Equation (6.5).  While this procedure may appear to be rather cumbersome, it provides a more efficient method of computer evaluation than the series formula, Equation (2.15).

 

            If we differentiate Equation (1.4) with respect to x, we get

 

                                                                                (6.8)

 

and, combining this with Equation (6.2), we obtain

 

                                                                                  (6.9)

 

from which a comparison of the coefficients of  yields the three-term recurrence relation

 

                                                                                    (6.10)

 

Differentiation of Equation (6.5) gives

 

                                       (6.11)

 

which, when combined with Equation (6.10) to eliminate , gives

 

          (6.12)

 

Upon simplification, this yields the recurrence relation

 

                                                                          (6.13)

 

Subtracting Equation (6.10) from Equation (6.13), we get

 

                                                                            (6.14)

 

 

            If we multiply Equation (6.10) by x, we get

 

                                                                                   (6.15)

 

and replacing n by  Equation (6.14) gives

 

                                                                                        (6.16)

 

Subtracting Equations (6.15) and (6.16), we get

 

                                                                      (6.17)

 

If we now differentiate Equation (6.17) with respect to x, we obtain the relation

 

                                                  (6.18)

 

The quantity in square brackets is obtained from Equation (6.10), to give

 

                                                             (6.19)

 

                                                        (6.20)

 

Thus we see that, if n is an integer, the Legendre polynomial  satisfies the differential equation

                                                          (6.21)

 

Equation (6.21) is known as Legendre’s differential equation.