ORTHOGONALITY OF BESSEL FUNCTIONS
19. ORTHOGONALITY OF BESSEL FUNCTIONS
You may expect here that we are going to prove that two J p ’s for different p values are orthogonal. However, this is not what we are going to do—as a matter of fact it isn’t true! To see what we are going to prove, look at the following comparison between Bessel functions and sines and cosines.
Two functions: sin x and cos x. Two functions for each p: J p (x) and N p (x).
Consider just sin x. Consider just J p (x) for one value of p.
At the zeros of sin x, namely, At the zeros of J p (x), say (19.1) x = nπ, sin x = 0.
x = α, β, · · · , J p (x) = 0. At x = 1, sin nπx = 0.
At x = 1, J p (αx) = 0, J p (βx) = 0, · · ·.
The differential equation satisfied The differential equation satisfied by y = sin nπx is
by y = J p (αx) is [see (16.5)] y ′′ + (nπ) 2 y = 0.
x(xy ′ ) ′ + (α 2 x 2 −p 2 )y = 0.
(In comparing the differential equations remember that p is a fixed constant. The correspondence is between the zeros of sin x, namely nπ, and the zeros of J p (x), namely, α, β, etc.)
We have proved (Chapter 7):
We shall prove:
sin nπx sin mπx dx = 0 xJ p (αx)J p (βx) dx = 0
By (16.5), the differential equation satisfied by J p (αx) is (19.2)
x(xy ′ ) ′ + (α 2 x 2 −p 2 )y = 0
and the differential equation satisfied by J p (βx) is (19.3)
x(xy ′ ) ′ + (β 2 x 2 −p 2 )y = 0.
Let us for simplicity call J p (αx) = u and J p (βx) = v; then (19.2) and (19.3) become
x(xu ′ ) ′ + (α 2 x 2 −p 2 )u = 0,
x(xv ′ ) ′ + (β 2 x 2 −p 2 )v = 0.
We are going to use equations (19.4) to prove the last equation in (19.1) by a method parallel to that used in proving the orthogonality of Legendre polynomials (Section 7). Multiply the first equation of (19.4) by v, the second by u, subtract the two equations and cancel an x to get
(19.5) v(xu ′ ) ′ − u(xv ′ ) ′ + (α 2 −β 2 )xuv = 0. The first two terms of (19.5) are equal to
(vxu ′ − uxv ′ ).
dx
602 Series Solutions of Differential Equations Chapter 12
Using (19.6) and integrating (19.5), we get
(vxu ′ − uxv ′ ) + (α 2 −β 2 )
xuv dx = 0.
At the lower limit the integrated term is zero because x = 0 and u, v, u ′ ,v ′ are finite. To evaluate the integrated term at the upper limit, recall that u = J p (αx), v=J p (βx); then at x = 1, u = J p (α) = 0, v = J p (β) = 0 since α and β are zeros of J p . The integrated term is therefore zero at the upper limit also. Thus (19.7) becomes
xJ p (αx)J p (βx) dx = 0.
p , the integral must be zero. If α = β, the integral is not zero; it can be evaluated, but we shall just state the answer (see Problem 1):
(19.10) xJ p (αx)J p (βx) dx =
0 1 J 2 (α) = 1 J 2 1 2 2 p+1 ′ 2 p−1 (α) = 2 J p (α) if α = β, where α and β are zeros of J p (x).
[You can see that the three answers for the case α = β are equal by equations (15.3) to (15.5), remembering that α is a zero of J p .]
We can state (19.10) in words in two different ways; if α n , n = 1, 2, 3, · · · , are the zeros of J p (x), then we say either that √ (a) the functions xJ p (α n x) are orthogonal on (0, 1); or that (b) the functions J p (α n x) are orthogonal on (0, 1) with respect to the weight function x. You may meet other sets of functions which are orthogonal with respect to a weight function. (See, for example, Section 22.) In general, we say that y n (x) is a set of orthogonal functions on (x 1 ,x 2 ) with respect to the weight function w(x) if
y n (x)y m (x)w(x) dx = 0
The fact that the Bessel functions J p (α n x) obey (19.10) makes it possible to expand a given function in a series of Bessel functions much as we expand functions in Fourier series and Legendre series. We shall do this later (Chapter 13) when we need it in a physical example.
Just as we generalized Fourier series to an interval (0, l), here we can generalize (19.10) to an interval (0, a). In (19.10), let x = r/a. Then the limits are x = r/a =
0 to 1, that is, r = 0 to a. The integral in (19.10) becomes
(r/a)J p (αr/a)J p (βr/a) d(r/a) =
rJ p (αr/a)J p (βr/a) dr.
Section 20 Orthogonality of Bessel Functions 603
Thus we have
(19.11) rJ p (αr/a)J p (βr/a) dr
J 2 (α) = a 2 p+1 2 2 J 2 p−1 (α) = a 2 2 2 J p ′ (α) if α = β.
PROBLEMS, SECTION 19
1. Prove equation (19.10) in the following way. First note that (19.2) and (19.3) and therefore (19.7) hold whether α and β are zeros of J p (x) or not. Let α be a zero, but let β be just any number. From (19.7) show that then
Z 1 J p (β)αJ p ′ (α)
xuv dx =
Now let β → α and evaluate the indeterminate form by L’Hˆopital’s rule (that is, differentiate numerator and denominator with respect to β and let β → α). Hence find
xuv dx = 1
0 2 J p ′ 2 (α)
for α = β, that is, for u = v = J p (αx) as in (19.10). Use equations (15.3) to (15.5) to show that the other two expressions given in (19.10) are equivalent.
2. Given that
„ sin x
x − cos x , use (19.10) to evaluate
where α is a root of the equation tan x = x. 3. Use (17.4) and (19.10) to write the orthogonality condition and the normalization
integral for the spherical Bessel functions j n (x). 4. Define J p (z) for complex z by the power series (12.9) with x replaced by z. (By
Problem 12.1, the series converges for all z.) Show by (19.10) that all the zeros of J p (z) are real. Hint: Suppose α and β in (19.10) were a complex conjugate pair; show that then the integrand would be positive so the integral could not be zero.
5. We obtained (19.10) for J p (x), p ≥ 0. It is, however, valid for p > −1, that is for N p (x), 0 ≤ p < 1. The difficulty in the proof occurs just after (19.7); we said that u, v, u ′ ,v ′ are finite at x = 0 which is not true for N p (x). However, the negative
powers of x cancel if p < 1. Show this for p = 1 2 by using two terms of the power
series (12.9) or (13.1) for the function N 1/2 (x) = −J −1/2 (x) [see (13.3)]. R 6. 1 By Problem 5,
0 xN 1/2 (αx)N 1/2 (βx) dx = 0 if α and β are different zeros of N 1/2 (x). Using (17.4), find N 1/2 (x) in terms of cos x and so find the zeros of N 1/2 (x). Show that the functions cos(n + 1 2 )πx are an orthogonal set on (0, 1). Use (19.10)
to find the normalization constant. (Compare Problem 6.8.)
604 Series Solutions of Differential Equations Chapter 12