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d 2 F (p) dF(p)
2--------+ cotp--------+ n(n + 1)F(p) = 0, 0 <p<n,
where n is a positive integer. Show that the change of variable x = cos p leads to the
Legendre equation with a = n for y = f (x) = F (arccos x).
27. Show that for n = 0, 1, 2, 3 the corresponding Legendre polynomial is given by
1 dn 2 n
pn(x) = d^(x 2 - 1)n.
This formula, known as Rodrigues (1794-1851) formula, is true for all positive integers n.
28. Show that the Legendre equation can also be written as
[(1 - x2)y']' = -a(a + 1)y.
Then it follows that [(1 x2)Pn(x)] = n(n + 1)Pn(x) and [(1 x2)P'm(x)]' = m (m + 1)Pm (x). By multiplying the first equation by Pm (x) and the second equation by P (x), and then integrating by parts, show that
Pn (x ) Pm (x ) dx = 0 if n = m.
This property of the Legendre polynomials is known as the orthogonality property. If m = n, it can be shown that the value of the preceding integral is 2/(2n + 1).
29. Given a polynomial f of degree n, it is possible to express f as a linear combination of
P P P P ?
r0 M ^2 ... rn'
f(x ) = J2 akpk(x ).
Using the result of Problem 28, show that
2k + 1 f1
j f (x)Pk(x) dx.
5.4 Regular Singular Points
ฎIn this section we will consider the equation
P (x) y" + Q(x) y + R (x) y = 0 (1)
in the neighborhood of a singular point x0. Recall that if the functions P, Q, and R are polynomials having no common factors, the singular points of Eq. (1) are the points for which P (x) = 0.
Chapter 5. Series Solutions of Second Order Linear Equations
Determine the singular points and ordinary points of the Bessel equation of order v
x 2y " + xy' + (x2 v 2)y = 0. (2)
The point x = 0 is a singular point since P (x) = x2 is zero there. All other points are ordinary points of Eq. (2).
Determine the singular points and ordinary points of the Legendre equation
(1 x 2)y" 2xy' + a(a + 1)y = 0, (3)
where a is a constant.
The singular points are the zeros of P (x) = 1 x2, namely, the points x = ฑ1. All other points are ordinary points.
Unfortunately, if we attempt to use the methods of the preceding two sections to solve Eq. (1) in the neighborhood of a singular point x0, we find that these methods fail. This is because the solution of Eq. (1) is often not analytic at x0, and consequently cannot be represented by a Taylor series in powers of x x0. Instead, we must use a more general series expansion.
Since the singular points of a differential equation are usually few in number, we might ask whether we can simply ignore them, especially since we already know how to construct solutions about ordinary points. However, this is not feasible because the singular points determine the principal features of the solution to a much larger extent than one might at first suspect. In the neighborhood of a singular point the solution often becomes large in magnitude or experiences rapid changes in magnitude. Thus the behavior of a physical system modeled by a differential equation frequently is most interesting in the neighborhood of a singular point. Often geometrical singularities in a physical problem, such as corners or sharp edges, lead to singular points in the corresponding differential equation. Thus, while at first we might want to avoid the few points where a differential equation is singular, it is precisely at these points that it is necessary to study the solution most carefully.
As an alternative to analytical methods, one can consider the use of numerical methods, which are discussed in Chapter 8. However, these methods are ill-suited for the study of solutions near a singular point. Thus, even if one adopts a numerical approach, it is advantageous to combine it with the analytical methods of this chapter in order to examine the behavior of solutions near singular points.
Without any additional information about the behavior of Q/P and R/P in the neighborhood of the singular point, it is impossible to describe the behavior of the solutions of Eq. (1) near x = x0. It may be that there are two linearly independent solutions of Eq. (1) that remain bounded as x ^ x0, or there may be only one with the other becoming unbounded as x ^ x0, or they may both become unbounded as x ^ x0. To illustrate these possibilities consider the following examples.
The differential equation
x2 y" 2 y = 0 (4)
has a singular point at x = 0. It can be easily verified by direct substitution that y1(x) = x2 and y2(x) = 1/x are linearly independent solutions of Eq. (4) for x > 0 or
5.4 Regular Singular Points
x < 0. Thus in any interval not containing the origin the general solution of Eq. (4) is y = c1 x2 + c2x1. The only solution of Eq. (4) that is bounded as x ^ 0 is y = c1 x2. Indeed, this solution is analytic at the origin even though if Eq. (4) is put in the standard form, y" (2/x2)y = 0, the function q (x) = 2/x2 is not analytic at x = 0, and Theorem 5.3.1 is not applicable. On the other hand, notice that the solution y2(x) = x1 does not have a Taylor series expansion about the origin (is not analytic at x = 0); therefore, the method of Section 5.2 would fail in this case.