12.6 Bessel and Legendre Series

INTRODUCTION

Fourier series, Fourier cosine series, and Fourier sine series are three ways of expanding a function in terms of an orthogonal set of functions. But such expansions are by no means limited to orthogonal sets of trigonometric functions. We saw in Section 12.1 that a function f defined on an interval (a, b) could be expanded, at least in a formal manner, in terms of any set of functions {øn(x)} that is orthogonal with respect to a weight function on [a, b]. Many of these orthogonal series expansions or generalized Fourier series derive from Sturm–Liouville problems that, in turn, arise from attempts to solve linear partial differential equations serving as models for physical systems. Fourier series and orthogonal series expansions (the latter includes the two series considered in this section) will appear in the subsequent consideration of these applications in Chapters 13 and 14.

12.6.1 Fourier–Bessel Series

We saw in Example 3 of Section 12.5 that for a fixed value of n the set of Bessel functions {Jn(αi x)}, i = 1, 2, 3, … , is orthogonal with respect to the weight function p(x) = x on an interval [0, b] when the αi are defined by means of a boundary condition of the form

(1)

The eigenvalues of the corresponding Sturm–Liouville problem are . From (7) and (8) of Section 12.1 the orthogonal series expansion or generalized Fourier series of a function f defined on the interval (0, b) in terms of this orthogonal set is

(2)

where (3)

The square norm of the function Jn(αi x) is defined by (11) of Section 12.1:

(4)

The series (2) with coefficients (3) is called a Fourier–Bessel series.

Differential Recurrence Relations

The differential recurrence relations that were given in (22) and (23) of Section 5.3 are often useful in the evaluation of the coefficients (3). For convenience we reproduce those relations here:

(5)

(6)

Square Norm

The value of the square norm (4) depends on how the eigenvalues λi = are defined. If y = Jn(αx), then we know from Example 3 of Section 12.5 that

After we multiply by 2xy′, this equation can be written as

Integrating the last result by parts on [0, b] then gives

Since y = Jn(αx), the lower limit is zero for n > 0 because Jn(0) = 0. For n = 0, the quantity [xy′]2 + α2x2y2 is zero at x = 0. Thus

(7)

where we have used the Chain Rule to write y′ = α(αx).

We now consider three cases of the boundary condition (1).

Case I: If we choose A2 = 1 and B2 = 0, then (1) is

Jn(αb) = 0.(8)

There are an infinite number of positive roots xi = αib of (8) (see Figure 5.3.1) that define the αi as αi = xi /b. The eigenvalues are positive and are then λi = /b2. No new eigenvalues result from the negative roots of (8) since Jn(–x) = (–1)nJn(x). (See page 295.) The number 0 is not an eigenvalue for any n since Jn(0) = 0 for n = 1, 2, 3, … and J0(0) = 1. In other words, if λ = 0, we get the trivial function (which is never an eigenfunction) for n = 1, 2, 3, … , and for n = 0, λ = 0 (or equivalently, α = 0) does not satisfy the equation in (8). When (6) is written in the form x (x) = n Jn(x) – x Jn+1(x), it follows from (7) and (8) that the square norm of Jn(αi x) is

(9)

Case II: If we choose A2 = h ≥ 0, B2 = b, then (1) is

(10)

Equation (10) has an infinite number of positive roots xi = αib for each positive integer n = 1, 2, 3, …. As before, the eigenvalues are obtained from λi = /b2. λ = 0 is not an eigenvalue for n = 1, 2, 3, …. Substituting αib(αib) = –hJn(αib) into (7), we find that the square norm of Jn(αi x) is now

(11)

Case III: If h = 0 and n = 0 in (10), the αi are defined from the roots of

(12)

Even though (12) is just a special case of (10), it is the only situation for which λ = 0 is an eigenvalue. To see this, observe that for n = 0, the result in (6) implies that (αb) = 0 is equivalent to J1(αb) = 0. Since x1 = αib = 0 is a root of the last equation, α1 = 0, and because J0(0) = 1 is nontrivial, we conclude from λ1 = that λ1 = 0 is an eigenvalue. But obviously we cannot use (11) when α1 = 0, h = 0, and n = 0. However, from the square norm (4) we have

(13)

For αi > 0 we can use (11) with h = 0 and n = 0:

(14)

The following definition summarizes three forms of the series (2) corresponding to the square norms in the three cases.

DEFINITION 12.6.1 Fourier–Bessel Series

The Fourier–Bessel series of a function f defined on the interval (0, b) is given by

(i) (15)

(16)

where the αi are defined by Jn(αb) = 0.

(ii) (17)

(18)

where the αi are defined by hJn(αb) + αb(αb) = 0.

(iii) (19)

(20)

where the αi are defined by (αb) = 0.

Convergence of a Fourier–Bessel Series

Sufficient conditions for the convergence of a Fourier–Bessel series are not particularly restrictive.

THEOREM 12.6.1 Conditions for Convergence

Let f and f′ be piecewise continuous on the interval [0, b]. Then for all x in the interval (0, b), the Fourier–Bessel series of f converges to f(x) at a point where f is continuous and to the average

at a point where f is discontinous.

EXAMPLE 1 Expansion in a Fourier–Bessel Series

Expand f(x) = x, 0 < x < 3, in a Fourier–Bessel series, using Bessel functions of order one that satisfy the boundary condition J1(3α) = 0.

SOLUTION

We use (15) where the coefficients ci are given by (16) with b = 3:

To evaluate this integral we let t = αi x, dx = dt/αi, x2 = t2/, and use (5) in the form [t2J2(t)] = t2J1(t):

Therefore the desired expansion is

≡

You are asked to find the first four values of the αi for the foregoing Bessel series in Problem 1 in Exercises 12.6.

EXAMPLE 2 Expansion in a Fourier–Bessel Series

If the αi in Example 1 are defined by J1(3α) + α(3α) = 0, then the only thing that changes in the expansion is the value of the square norm. Multiplying the boundary condition by 3 gives 3 J1(3α) + 3α(3α) = 0, which now matches (10) when h = 3, b = 3, and n = 1. Thus (18) and (17) yield, in turn,

and ≡

Use of Computers

Since Bessel functions are “built-in functions” in a CAS, it is a straightforward task to find the approximate values of the αi and the coefficients ci in a Fourier–Bessel series. For example, in (9) we can think of xi = αib as a positive root of the equation h Jn(x) + x (x) = 0. Thus in Example 2 we have used a CAS to find the first five positive roots xi of 3J1(x) + x (x) = 0 and from these roots we obtain the first five values of αi: α1 = x1 /3 = 0.98320, α2 = x2/3 = 1.94704, α3 = x3/3 = 2.95758, α4 = x4 /3 = 3.98538, and α5 = x5/3 = 5.02078. Knowing the roots xi = 3αi and the αi , we again use a CAS to calculate the numerical values of J2(3αi), (3αi), and finally the coefficients ci. In this manner we find that the fifth partial sum S5(x) for the Fourier–Bessel series representation of f(x) = x, 0 < x < 3 in Example 2 is

S5(x) = 4.01844 J1(0.98320x) – 1.86937 J1(1.94704x)

+ 1.07106 J1(2.95758x) – 0.70306 J1(3.98538x) + 0.50343 J1(5.02078x).

The graph of S5(x) on the interval (0, 3) is shown in FIGURE 12.6.1(a). In FIGURE 12.6.1(b) we have graphed S10(x) on the interval (0, 50). Notice that outside the interval of definition (0, 3) the series does not converge to a periodic extension of f because Bessel functions are not periodic functions. See Problems 11 and 12 in Exercises 12.6.

Two graphs of Partial sums of a Fourier–Bessel series. The first graph captioned (a) S subscript 5 (x), 0 less than x less than 3 has a curve graphed on an x y plane. The curve starts at the point (0, 0), goes up and to the right with very slight oscillation, and ends at the point (3, 2.5). The second graph captioned (b) S subscript 10 (x), 0 less than x less than 50 has a curve graphed on an x y plane. The curve starts at the approximate point (0, 0.2), goes up and to the right, reaches a high point, then goes down and to the right through the positive x axis, reaches a low point, again goes up and to the right through the positive x axis, follows an oscillatory pattern, and ends at the approximate point (50, negative 0.5).

FIGURE 12.6.1 Partial sums of a Fourier–Bessel series

12.6.2 Fourier–Legendre Series

From Example 4 of Section 12.5 we know that the set of Legendre polynomials {Pn(x)}, n = 0, 1, 2, …, is orthogonal with respect to the weight function p(x) = 1 on the interval [–1, 1]. Furthermore, it can be proved that the square norm of a polynomial Pn(x) depends on n in the following manner:

The orthogonal series expansion of a function in terms of the Legendre polynomials is summarized in the next definition.

DEFINITION 12.6.2 Fourier–Legendre Series

The Fourier–Legendre series of a function f defined on the interval (–1, 1) is given by

(21)

where (22)

Convergence of a Fourier–Legendre Series

Sufficient conditions for convergence of a Fourier–Legendre series are given in the next theorem.

THEOREM 12.6.2 Conditions for Convergence

Let f and f′ be piecewise continuous on the interval [–1, 1]. Then for all x in the interval (–1, 1), the Fourier–Legendre series of f converges to f(x) at a point where f is continuous and to the average

at a point where f is discontinuous.

EXAMPLE 3 Expansion in a Fourier–Legendre Series

Write out the first four nonzero terms in the Fourier–Legendre expansion of

SOLUTION

The first several Legendre polynomials are listed on page 299. From these and (22) we find

Hence ≡

Like the Bessel functions, Legendre polynomials are built-in functions in computer algebra systems such as Mathematica and Maple, and so each of the coefficients just listed can be found using the integration application of such a program. Indeed, using a CAS, we further find that c6 = 0 and c7 = –. The fifth partial sum of the Fourier–Legendre series representation of the function f defined in Example 3 is then

The graph of S5(x) on the interval (–1, 1) is given in FIGURE 12.6.2.

A curve is graphed on an x y plane. It starts at the approximate point (negative 1, 0.1), goes down and to the right through the positive x axis, reaches a point in the third quadrant, then, goes up and to the right, and follows an oscillatory pattern to reach a point in the first quadrant through the positive y axis. Then it goes up and down and to the right, and ends at the approximate point (1, 0.9).

FIGURE 12.6.2 Partial sum S5(x) of Fourier–Legendre series in Example 3

Alternative Form of Series

In applications, the Fourier–Legendre series appears in an alternative form. If we let x = cos θ, then x = 1 implies θ = 0, whereas x = –1 implies θ = π. Since dx = –sin θ dθ, (21) and (22) become, respectively,

(23)

(24)

where f(cos θ) has been replaced by F(θ).

12.6 Exercises Answers to selected odd-numbered problems begin on page ANS-33.

12.6.1 Fourier–Bessel Series

In Problems 1 and 2, use Table 5.3.1 in Section 5.3.

  1. Find the first four αi > 0 defined by J1(3α) = 0.
  2. Find the first four αi ≥ 0 defined by (2α) = 0.

In Problems 3–6, expand f(x) = 1, 0 < x < 2, in a Fourier–Bessel series using Bessel functions of order zero that satisfy the given boundary condition.

  1. J0(2α) = 0
  2. (2α) = 0
  3. J0(2α) + 2α (2α) = 0
  4. J0(2α) + α (2α) = 0

In Problems 7–10, expand the given function in a Fourier–Bessel series using Bessel functions of the same order as in the indicated boundary condition.

  1. f(x) = 5x, 0 < x < 4
    3J1(4α) + 4α (4α) = 0
  2. f(x) = x2, 0 < x < 1
    J2(α) = 0
  3. f(x) = x2, 0 < x < 3
    (3α) = 0
    [Hint: t3 = t2 ⋅ t.]
  4. f(x) = 1 – x2, 0 < x < 1
    J0(α) = 0

Computer Lab Assignments

    1. Use a CAS to graph y = 3J1(x) + x(x) on an interval so that the first five positive x-intercepts of the graph are shown.
    2. Use the root-finding capability of your CAS to approximate the first five roots xi of the equation

      3J1(x) + x (x) = 0.

    3. Use the data obtained in part (b) to find the first five positive values of αi that satisfy

      3J1(4α) + 4α (4α) = 0.

      See Problem 7.

    4. If instructed, find the first 10 positive values of αi.
    1. Use the values of αi in part (c) of Problem 11 and a CAS to approximate the values of the first five coefficients ci of the Fourier–Bessel series obtained in Problem 7.
    2. Use a CAS to graph the partial sums SN (x), N = 1, 2, 3, 4, 5, of the Fourier–Bessel series in Problem 7.
    3. If instructed, graph the partial sum S10(x) for 0 < x < 4 and for 0 < x < 50.

Discussion Problems

  1. If the partial sums in Problem 12 are plotted on a symmetric interval such as (–30, 30), would the graphs possess any symmetry? Explain.
    1. Sketch, by hand, a graph of what you think the Fourier–Bessel series in Problem 3 converges to on the interval (–2, 2).
    2. Sketch, by hand, a graph of what you think the Fourier–Bessel series would converge to on the interval (–4, 4) if the values αi in Problem 7 were defined by 3J2(4α) + 4α(4α) = 0.

12.6.2 Fourier–Legendre Series

In Problems 15 and 16, write out the first five nonzero terms in the Fourier–Legendre expansion of the given function. If instructed, use a CAS as an aid in evaluating the coefficients. Use a CAS to graph the partial sum S5(x).

  2. f(x) = ex, –1 < x < 1
  3. The first three Legendre polynomials are P0(x) = 1, P1(x) = x, and P2(x) = (3x2 – 1). If x = cos θ, then P0(cos θ) = 1 and P1(cos θ) = cos θ. Show that P2(cos θ) = (3 cos 2θ + 1).
  4. Use the results of Problem 17 to find a Fourier–Legendre expansion (23) of F(θ) = 1 – cos 2θ.
  5. A Legendre polynomial Pn(x) is an even or odd function, depending on whether n is even or odd. Show that if f is an even function on the interval (–1, 1), then (21) and (22) become, respectively,

    (25)

    (26)

  6. Show that if f is an odd function on the interval (–1, 1), then (21) and (22) become, respectively,

    (27)

    (28)

The series (25) and (27) can also be used when f is defined on only the interval (0, 1). Both series represent f on (0, 1); but on the interval (–1, 0), (25) represents an even extension, whereas (27) represents an odd extension. In Problems 21 and 22, write out the first four nonzero terms in the indicated expansion of the given function. What function does the series represent on the interval (–1, 1)? Use a CAS to graph the partial sum S4(x).

  1. f(x) = x, 0 < x < 1; (25)
  2. f(x) = 1, 0 < x < 1; (27)

Discussion Problems

  1. Why is a Fourier–Legendre expansion of a polynomial function that is defined on the interval (–1, 1) necessarily a finite series?
  2. Use your conclusion from Problem 23 to find the finite Fourier–Legendre series of f(x) = x2. The series of f(x) = x3. Do not use (21) and (22).