15.1.1 Series solutions about a regular point

With the regular point taken to be x = 0, (15.3a) becomes

(15.4a)

so that

(15.4b)

and

(15.4c)

If we now substitute (15.4) into the general second-order linear equation

we can extract the coefficient of each power of x in terms of the coefficients a_n. For a general solution, the coefficient of each power of x must be equal on both sides of the equation, and so each coefficient must separately vanish. This leads to relations between the various coefficients a_n. These are called recurrence relations and allow values of a_n for higher values of n to be found from values of a_n for lower values of n. This is best illustrated by an example.

We will find the series solution of the equation y′′ + 7y = 0 about the point x = 0. Firstly, by inspection, x = 0 is a regular point and we can use the results (15.4). Substituting these into y′′ + y = 0, gives

and since each term must vanish separately, we obtain the two-term recurrence relation

Thus all the even coefficients may be found in terms of a₀ and all the odd coefficients may be found in terms of a₁. In the first case,

and in the second case,

The first result can be recognised as the series for and the second that for , so the general solution is

where a₀ and a₁ are arbitrary constants that would have to be fixed by suitable boundary conditions.

In this example, the solution is expressible in closed form and so may also have been found by the (easier) techniques described in Chapter 14, but it is useful to illustrate the general method. The series method is most useful for cases where no closed form for the solution exists. The following is an example of such a case.

15.1.2 Series solutions about a regular singularity: Frobenius method

If we wish to expand about a singular point x₀, we use a method due to Frobenius starting from the series (15.3b). Assuming again that x₀ = 0, this becomes

(15.5)

where c is not necessarily integer and a₀ ≠ 0. As x = 0 is a regular singularity, we can define new functions s(x) and t(x) by s(x) ≡ xp(x) and t(x) ≡ x²q(x), both of which have simple expansions

(15.6)

The original ODE may now be written in terms of these new functions as

(15.7)

where the derivatives are

(15.8a)

and

(15.8b)

Substituting (15.8) into (15.7), gives

(15.9)

The coefficient of the lowest power of x, that is, x^{c − 2} is, from (15.9) and (15.6),

and must vanish by (15.5), so that

(15.10)

This is called the indicial equation. It is a quadratic in c with two roots c₁ and c₂, called the indices of the regular singular point. Each solution, when used in (15.9), and requiring the coefficients to all vanish separately, leads to a recurrence relation between the a_n and hence to a solution of the original ODE. Again, this is best illustrated by an example.We will find the power series solution of

about the point x = 0. In the standard notation used previously and , and it is straightforward to show that x = 0 is a regular singular point. Therefore using the Frobenius series (15.5) and (15.8) in the above equation gives an analogous equation to (15.9), i.e.,

Setting the coefficient of the lowest power of x to zero, we obtain the indicial equation

with roots . Demanding that the coefficients of each power x vanish separately gives the recurrence relation

(15.11)

Consider firstly the case, . The recurrence relation (15.11) becomes

Setting a₀ = 1, we can calculate and from this , , etc. The corresponding solution of (15.10) is

Similarly, for the second root c = 0, we find , , , etc., and the corresponding second solution

The general solution is therefore

where c₁ and c₂ are constants. As in Example 15.2, this solution is not in closed form, but the series again converge for all finite values of x.

In the above examples, the solutions obtained from each of the two roots of the indicial equation are linearly independent. While this is usually true, there are circumstances where it is not. An obvious example is when the two roots are equal. A second example is when the two indices differ by an integer. In this case, the recurrence relation may, or may not, lead to a second solution that is linearly independent. To illustrate this we will find a power series solution of the equation

about the point x = 0. Using the previous notations,

and so x = 0 is a regular singular point. Proceeding as above, using the expansions (15.8), leads to an equation analogous to (15.9), i.e.

If we now multiply throughout by (x − 1), we have

(15.12)

and setting the coefficient of the lowest power of x to zero, that is, the coefficient of x^{c − 2}, gives the indicial equation

with roots c = 0, 1. It can be shown that the larger root will always give a Frobenius solution.³ This is found by using c = 1 in (15.12) and setting the coefficient of each power of x to zero, giving

and hence the recurrence relation is

So setting a₀ = 1, gives a₁ = 3, a₂ = 6, etc. and hence

In the present example, the smaller root does not result in another power series solution, because repeating the procedure above for c = 0, we find the recurrence relation

and since we require a₀ ≠ 0, a₁ is infinite and the method fails.

In cases such as these, and those where the roots are equal, the Frobenius method yields a single series solution specified in terms of a single free parameter a₀. Since the general solution of a linear second order differential equation always depends on two free parameters, we need another method for finding a second independent solution. There are several ways of doing this. One is to use another result of Fuchs' theorem.⁴ This states that if y₁(x) is a Frobenius series, then a second solution is

(15.13a)

where z(x) has the Frobenius form

and d is the smaller of the roots of the original indicial equation. In general, the method is used by substituting y₂(x) into the original differential equation and finding a solution for b_n, with b₀ ≠ 0.

Alternatively, a more general method, which applies to any second-order linear equation where a solution y₁(x) is known, is to substitute

(15.13b)

into the differential equation and solve for u(x). It is illustrated in Example 15.4. Both these methods, and others, for finding a second solution are easiest to apply if the first solution is in a simple closed form.

15.1.3 Polynomial solutions

Another special class of solutions using the series method is when for some value n the coefficient a_n in the recurrence relation is zero. In this case, all subsequent coefficients generated from the recurrence relation will also be zero and the infinite series actually terminates at some finite n. The solutions are then finite-order polynomials and these polynomial solutions often have a special importance in physics.As an example, consider Hermite's equation

(15.14)

where λ is a constant parameter. We can easily see that x = 0 is a regular point and so an expansion about this point is

Substituting into the differential equation and proceeding as in Section 15.1.2, leads to the recurrence relation

Thus the even and odd coefficients are independent of each other; the even coefficients are given in terms of a₀ and the odd coefficients are given in terms of a₁. If we set a₀ = 1 and a₁ = 0, we obtain the solution

(15.15a)

while if we set a₀ = 0 and a₁ = 1 we obtain a second solution

(15.15b)

The general solution is then

where A and B are arbitrary constants.

The solutions (15.15) are, in general, infinite series. To obtain a polynomial solution, we must set λ = 2k, where k > 0 is an integer, so that the recurrence relation gives a_{2 + k} = 0, and one of the series (15.15a) and (15.15b) terminates. If k is even and we set a₀ = 1, a₁ = 0, the series (15.15a) terminates, giving a polynomial solution h_k(x) of order k. For example,

and so on. Alternatively, If k is odd and we set a₀ = 0, a₁ = 1, the series (15.15b) terminates, again giving a polynomial solution h_k(x) of order k. For example,

Any polynomial of the form H_k(x) = c_kh_k(x), where the c_k are constants, is called a Hermite polynomial. The convention for choosing the c_k is not universal, but in physics they are chosen so that the coefficient of x^k in H_k(x) is 2^k, and the first six polynomials are then:

These polynomials occur in the quantum mechanical theory of the simple harmonic oscillator.

15.3.1 Legendre functions and Legendre polynomials

The lack of convergence at x = ±1 of the series obtained using (15.26) is an important limitation, because in many physics applications, particularly those in quantum theory, x is the cosine of an angle and l is a non-negative integer. Thus we need to find solutions that converges for all x, including x = ±1. This is only possible for integer values of l, as we shall show below.

The general solution of (15.23) is the sum of two series containing two constants a₀ and a₁. Using the recurrence relation (15.26) we may therefore write

(15.27a)

Now if, and only if, l is a non-negative integer, one of these series will terminate at l = n and the other will diverge at x = ±1. This is simply seen by considering the series for l = 0 at x = 1. In this case, the even solution is simply a₀ and the odd series is

which diverges. However, if l = 1, the odd series is just a₁x, whereas the even series diverges at x = ±1.

The series that terminates defines a finite polynomial of order l, called a Legendre polynomial and written P_l(x). The other series diverges at x = ±1 and defines a Legendre function of the second kind, written Q_l(x). For integer l, the general solution of the Legendre equation is then

(15.27b)

The functions Q_l(x) occur far less frequently in physical applications than the polynomials and we will therefore focus mainly on the latter functions. From (15.27a), if we choose the value of either a₀ or a₁ so that y_l(1) = 1, and hence y_l( − 1) = ( − 1)^l, then the first three even-order polynomials are

(15.28a)

and the first three odd polynomials are

(15.28b)

Choosing the constants in this way ensures that the polynomials satisfy the normalisation condition

(15.29a)

while the odd and even powers in the series imply

(15.29b)

The first four Legendre polynomials are plotted in Figure 15.1a. The polynomial of order l in general has l nodes, and as l increases the polynomials oscillate more and more rapidly, as illustrated in Figure 15.1b for l = 10.

The Legendre polynomials satisfy the orthogonality relation

(15.30)

where δ_lm is the Kronecker delta symbol (9.24b). For l = m, this reduces to

(15.31)

and is a consequence of the normalisation convention (15.29a). It may be verified for individual cases using (15.28) and will be proved in general in the next section. For l ≠ m, (15.30) may be proved by starting from the Legendre equation (15.23), which is conveniently rewritten in the form

Setting y(x) = P_l(x), and writing this equation for two values l and m, gives

and

Multiplying the first of these by P_m(x) and the second by P_l(x), and then subtracting one equation from the other gives

Then integrating both sides over x from –1 to +1, we have

The left-hand side of this equation may be shown to vanish by integrating both terms by parts, and it follows that if l ≠ m,

as required.

The orthogonality relation (15.30) is often used in conjunction with another result, which we will state without proof. This is that any function f(x) that is non-singular in the range − 1 ≤ x ≤ 1 can be expanded in a convergent series of the form

(15.32)

This property is called completeness and P_l(x), l = 0, 1, 2, …, are called a complete set of functions, in analogy to the definition of a complete set of basis vectors in Section 9.2.1. On multiplying (15.32) by P_n(x) and integrating, one obtains

(15.33)

for the coefficients in (15.32). This expansion is called a Legendre series and is closely analogous to a Fourier expansion, as can be seen by comparing (15.30), (15.32) and (15.33) with (13.38), (13.39a) and (13.39b) respectively.⁸ The expansion (5.32) is often used in numerical work where one has a large number of measurements of a quantity f as a function of angle, that is, an angular distribution f(cos θ), and requires a convenient approximate representation of them.

We conclude this section with a brief account of the Legendre functions of the second kind. As discussed earlier, these are defined by the first series in (15.27a) for odd l, where by convention we take a₀ = 1; and by the second series in (15.27a) for even l, where we take a₁ = 1. For integer l, the resulting series can be conveniently summarised by introducing the double factorial

(15.34a)

which satisfy the identities

(15.34b)

where 0!! = 1, by definition. Using relations like

one finds from (15.27a) and (15.27b) that

(15.35)

These series diverge at x = ±1, as shown by expressing them in closed form for l = 0, 1 in Example 15.8, and generalising the result to all integer l in Problem 15.14.

*15.3.2 The generating function

A useful technique for deriving properties of Legendre polynomials is to use the generating function

(15.36)

where h is a dummy variable and

(15.37)

To prove (15.37), we have to show that the functions P_l(x) on the right-hand side really do satisfy the Legendre equation and that they have the property P_l(1) = 1. The latter follows simply by putting x = 1 in (15.36) so that

and then equating this to the right-hand side of (15.37) to give

Since this relation is an identity in h, the coefficients of hⁿ on both sides must be equal and so P_l(1) = 1. To show that the P_l(x) in (15.37) satisfy the Legendre equation, we use the identity

(15.38)

that may be verified from the definition (15.36). Substituting (15.37) into (15.38) gives

Since this is an identity in h, the coefficient of each power of h must vanish, and hence

But this is the Legendre equation, and so the P_l of (15.37) are indeed Legendre functions.

The generating function is useful in deriving recurrence relations for Legendre polynomials. These are relations that relate two or more polynomials of different orders, that is, with different values of l, and by analogy to the recurrence relations discussed earlier, they provide a simple way of evaluating higher-order polynomials from polynomials of lower order.⁹ Some examples of recurrence relations are:

(15.39a)

(15.39b)

(15.39c)

(15.39d)

(15.39e)

As an example of how these are derived using the generating function, we will prove (15.39b). Differentiating (15.36) and (15.37) partially with respect to x, keeping h constant, gives

while differentiating with respect to h, keeping x constant, gives

Comparing these two equations gives

and equating the coefficients of h^l gives

which is (15.39b). Proofs of some of the other relations are left to the Examples and Problems. One can show that exactly the same recurrence relations apply to the Legendre functions of the second kind (see Problem 15.12).

The generating function also yields an elegant derivation of the normalisation formula (15.31). To do this, we evaluate

using (15.36) and (15.37). From (15.36) we obtain

(15.40a)

where we have used the Maclaurin expansion of Table 5.1 to expand the logarithms. On the other hand, using (15.37) gives

(15.40b)

where we have used the orthogonality relation (15.33). Equating powers of 2l in (15.40a) and (15.40b) yields (15.31) as required.

Finally, a well-known physical application of (15.36) and (15.37) is the expansion of a potential V(r) due to a point charge, or mass, at r = a, in powers of , where r = |r|. From Figure 15.2, we have in the electrostatic case

where q is the charge and ϵ₀ is the permittivity of the vacuum. Writing, a = |a|, and , for r > a we have

and expanding this using (15.36) and (15.37) gives

(15.41a)

For r < a, the corresponding result is

(15.41b)

Equations (15.41) are called the axial multipole expansions. Using them, the potential due to any linear distribution of point charges can then be obtained by adding the contributions of each point charge using (15.41). For example, for a dipole with − e at r = 0 and + e at r = a, one obtains the simple result

in the limit r ≫ a, where μ = ea is the dipole moment.

*15.3.3 Associated Legendre equation

Another equation that is closely associated with the Legendre equation is the associated Legendre equation

(15.42)

where m and l are integers and in physical situations − l ≤ m ≤ l. This equation reduces to the Legendre equation if m = 0, but in physical applications it is often the family of equations (15.42) that occurs, rather than just the Legendre equation itself. However, the solutions of (15.42), called the associated Legendre functions, are easily obtained from the Legendre functions already derived, as we now show.

To do this, we substitute

(15.43)

into (15.42) to obtain, after some simplification,

(15.44)

On the other hand, on differentiating Legendre's equation (15.23) m times, we obtain

(15.45)

Comparing (15.44) and (15.45), we see that , where y is a solution of Legendre's equation. Hence from (15.42) and (15.27b) the general equation for m ≥ 0 is

(15.46)

In applications, we are mostly interested in the associated Legendre polynomials

(15.47a)

and since the associated Legendre equation depends only on m², we can define

where c_lm is a constant. The usual convention is to define (cf. Section 15.3.4 below)

(15.47b)

when the orthogonality relation analogous to (15.30) for given m is¹⁰

(15.48)

*15.3.4 Rodrigues' formula

In the previous sections we have derived the properties of Legendre polynomials from the properties of Legendre's equation, or by using the generating function (15.36) and (15.37). An alternative approach is to exploit, or even define, the polynomials by using Rodrigues' formula,

(15.49)

To derive this result, we note that for even l, the Legendre polynomials can be written in the compact form [cf. Problems (15.12) and (15.13)]

(15.50a)

Since

this becomes

(15.51)

where the sum has been extended to all k ≤ l. The reason for this is that we can now use the binomial theorem (1.23) and (1.24), to write

and Rodrigues' formula (15.49) follows. A similar argument, starting from the expansion

(15.50b)

establishes Rodrigues' formula for odd l also.

Rodrigues' formula can be used to derive many useful results on Legendre polynomials as illustrated in Example 15.10. It is also easily extended to associated Legendre polynomials by substituting (15.49) into (15.47a) to give

(15.52)

for m ≥ 0. However, although this formula is derived from (15.47a) for m ≥ 0, the right-hand side is defined for negative m ≥ −l; and if it is used to define P^{− m}_l(x), it can be shown to automatically lead to the normalisation (15.47b) for P^{− m}_l(x) adopted in the previous section.

15.4.1 Bessel functions

We start by considering the case where c = ν. From (15.58),

or, equivalently,

(15.59)

Using this recurrence relation we find

(15.60)

and so on. If ν is a positive integer, it can be seen that the denominator can be written compactly in terms of factorials, but in the general case where ν is not an integer, we need to use a notation that reduces to factorials for integral ν. The required function is called a gamma function Γ(ν) and is defined for positive ν by

(15.61)

It can be shown from this definition, by integrating by parts (see Problem 4.12), that

(15.62a)

This is a recurrence relation for the gamma function and can be used, together with (15.61), to extend the definition from ν > 0 to all ν, including ν ≤ 0. For integer n ≥ 0, together with Γ(1) = 1 obtained directly from (15.61), it leads to

(15.62b)

with 0! ≡ Γ(1) = 1, while for integers n ≤ 0 one has

(15.62c)

where the sign depends on the direction of approach to the limit. The resulting behaviour of the gamma function for − 5 ≤ ν ≤ 4 is shown in Figure 15.3.

Returning to the series defined by (15.59), we see that the relations (15.60), written in terms of gamma functions, are

and in general,

(15.63)

It is usual to set

(15.64)

and the function y(x) is then called the Bessel function of the first kind of order ν, written J_ν(x). Using (15.54), (15.62), (15.63) and (15.64), we find

(15.65)

We next consider the case where c = −ν. It is not necessary to repeat all the steps that led to the derivation of (15.65). All we have to do is replace ν by − ν in that equation. This gives

(15.66)

The series (15.65) and (15.66) are easily shown to converge for 0 < x < ∞ using the ratio test and J_{− ν}(x), like J_ν(x), is also called a Bessel function of the first kind.

At this point, we distinguish between integer and non-integer ν. For non-integer ν, J_ν(x) and J_{− ν}(x) are independent solutions, as is the linear combination

(15.67)

where c₁ and c₂ are arbitrary constants. However, as can be seen from the first terms in (15.65) and (15.66), only J_ν(x) with ν ≥ 0 is non-singular as x → 0.

For integer ν = m > 0, the situation is somewhat different. This is because the first terms in (15.66) vanish by (15.62c), so that

where we have defined k = n − m. Hence for integer m, J_m(x) and J_{− m}(x) are not independent solutions and another solution must be found. For this reason, it is conventional to replace J_{− ν}(x) by the function

(15.68)

These functions are called Bessel functions of the second kind.¹¹ For non-integer ν, they are obviously solutions of Bessel's equation, since they are just well-defined linear combinations of J_ν(x) and J_{− ν}(x). However, it can be shown that they are also solutions for integer m, provided we interpret (15.68) as

(15.69)

The general solution of Bessel's equation is then written

(15.70)

for both integer and non-integer ν, where A and B are arbitrary constants.

We will not discuss the functions N_ν(x) further, because only Bessel functions of the first kind J_ν(x) with ν ≥ 0 are non-singular as x → 0. These, and especially those with integer ν, are the most important in applications, and the behaviour of J_n(x) are shown in Figure 15.4 for n = 0, 1, 2 and 3, and 0 ≤ x ≤ 10. As seen from (15.65), J_n(0) = 0 for n > 0. The positions of the zeros for x > 0 are also important in applications. The values of the first five zeros of the Bessel functions J_n(x), n = 1, 2, …, 5 are given in Table 15.1.

Further properties of Bessel functions that are useful in applications are discussed in the next subsection. However, before doing so, we warn the reader that J_ν(x) and N_ν(x) are not the only forms referred to as Bessel functions. There are others, such as spherical Bessel functions (that arise in scattering problems) and Hankel functions. We will not discuss these other forms here.

*15.4.2 Properties of non-singular Bessel functions J_ν(x)

The values and properties of the various types of Bessel functions are extensively listed in reference books and on the web.¹² Here we restrict ourselves to just some of the properties of Bessel functions that are non-singular at x = 0, that is, Bessel functions of the first kind J_ν(x) (ν > 0).

Bessel functions obey recurrence relations that are somewhat similar to those obtained in Section 15.3.2 for Legendre polynomials. Some of these recurrence relations, which hold for positive and negative ν, are

(15.71a)

(15.71b)

(15.71c)

(15.71d)

(15.71e)

Such relations are easily confirmed using the series representation (15.65). For example, if we differentiate the product x^νJ_ν(x) using (15.65), we obtain

which is (15.71a). Expanding the left-hand side of this expression and dividing by x^{ν − 1}, gives

(15.72a)

In a similar way we may show that

(15.72b)

These relations are equivalent to (15.71e), and adding them and dividing by x gives (15.71d).

In Section 15.3.1, we saw an arbitrary function that is non-singular in the range − 1 < x < 1 could be expanded in terms of Legendre polynomials [cf. (15.32)]. If a_νn > 0 are the zeroes of the Bessel function J_v(x), i.e.

(15.73)

a similar expansion in terms of Bessel functions in the range 0 < x < 1 can be obtained by considering the functions J_ν(a_νnx), which satisfy the relations¹³

(15.74)

For m ≠ n, this differs from the orthogonality relations (15.33) obtained for Legendre polynomials by the presence of the factor x and in the range of integration. For this reason, the functions J_ν(a_νnx) are said to be orthogonal with weight function x in the domain 0 ≤ x ≤ 1. In analogy to (15.32), it can be shown that an arbitrary function f(x) that is non-singular in the domain 0 ≤ x ≤ 1 can be expanded in the form

(15.75)

i.e.J_ν(a_νnx), n = 1, 2, 3, …, form a complete set of functions in this range. The coefficients c_νk are then obtained by multiplying (15.75) by xJ_ν(a_νkx) and integrating using (15.74) to give

(15.76)

The expansion (15.75) is called a Fourier-Bessel series and is often used in the solution of partial differential equations in cylindrical polar co-ordinates.