OTHER KINDS OF BESSEL FUNCTIONS

17. OTHER KINDS OF BESSEL FUNCTIONS

We have discussed J p (x) and N p (x) which are called Bessel functions of the first and second kinds , respectively. Since Bessel’s equation is of second order, there are, of course, only two independent solutions. However, there are a number of related functions which are also called Bessel functions. Here again there is a close analogy to sines and cosines. We may think of cos x and sin x as the solutions of y ′′ + y = 0. But cos x ± i sin x are also solutions which we usually write as e ±ix . If we replace x by ix, we get the functions e x ,e −x , cosh x, sinh x, which are solutions of y ′′ − y = 0. We list a number of Bessel functions which are frequently used and their trigonometric analogues:

Hankel Functions or Bessel Functions of the Third Kind

H p (1) (x) = J p (x) + iN p (x),

H p (2) (x) = J p (x) − iN p (x).

(Compare e ±ix = cos x ± i sin x.)

Modified or Hyperbolic Bessel Functions The solutions of (17.2)

x 2 y ′′ + xy ′

2 +p − (x 2 )y = 0

are, by (16.1) Z p (ix). (Compare this with the standard Bessel equation and by analogy consider the relation between y ′′ + y = 0 and y ′′ − y = 0.) The two independent solutions of (17.2) which are ordinarily used are

I p (x) = i −p J p (ix),

2 p (ix).

K p (x) = i p+1 H (1)

These should be compared with sinh x = −i sin(ix) and cosh x = cos(ix); because of the analogy, I and K are called hyperbolic Bessel functions. The i factors are adjusted to make I and K real for real x.

596 Series Solutions of Differential Equations Chapter 12

Spherical Bessel Functions If p = (2n + 1)/2 = n + 1 2 , n an integer, then J p (x) and N p (x) are called Bessel functions of half-odd integral order; they can be expressed in terms of sin x, cos x, and powers of x. The spherical Bessel functions are closely related to them as you can see from the formulas (17.4) below. Spherical Bessel functions arise in a variety of vibration problems especially when spheri- cal coordinates are used. We define the spherical Bessel functions j n (x), y n (x),

h (2) n (x), h n (x), for n = 0, 1, 2, · · · , and state their values in terms of elementary functions (see Problems 2 and 3). For the use of these functions, see Chapter 13, Problems 7.15, 7.16, 7.19, and 10.20.

j n (x) =

2x (2n+1)/2

1 d cos x (17.4)

Y (2n+1)/2 (x) = −x n

2x

x dx

h (1) n =j n (x) + iy n (x),

h (2) n =j n (x) − iy n (x).

A standard method of solving vibration problems is to as- sume a solution involving e iωt ; the resulting equation may contain imaginary terms. As an example, the following equation arises in the problem of the distribution of alternating current in wires (skin effect) (Relton, p. 177):

The Kelvin Functions

The solution of this equation is (Problem 8a) (17.6)

y=Z (i 3/2 0 x).

This is complex, and it is customary to separate it into its real and imaginary parts, called (for Z = J) ber and bei; these stand for Bessel-real and Bessel-imaginary. We define the ber, bei, ker, kei functions by

heat flow and in the theory of viscous fluids, as well in electrical engineering. The Airy Functions The Airy differential equation is

y ′′ − xy = 0.

By Section 16, the solutions are (Problem 8b)

xZ 1/3 ( 2 3 ix 3/2 ),

Section 17 Other Kinds of Bessel Functions 597

so by (17.3) they can be written in terms of I 1/3 and K 1/3 . The Airy functions are defined as

Ai(x) =

3 1/3 3 x

(17.10) Bi(x) = I 2 3/2 2 3/2

. For negative x, Ai and Bi can be expressed in terms of J 1/3 and N 1/3 , or the Hankel

3 −1/3 3 x

+I 1/3 3 x

functions (17.1) of order 1/3. Airy functions are of use in electrodynamics and quantum mechanics.

PROBLEMS, SECTION 17

1. Write the solutions of Problem 16.1 as spherical Bessel functions using the definitions (17.4) of j n (x) and y n (x) in terms of J (2n+1)/2 (x) and Y (2n+1)/2 (x). Then, using (17.4), obtain the solutions in terms of sin x and cos x. Compare with the answers in equation (11.6) and Problem 11.1.

2. From Problem 12.9, J 1/2 (x) = p2/πx sin x. Use (15.2) to obtain J 3/2 (x) and J 5/2 (x). Substitute your results for the J’s into (17.4) to verify the formulas stated

for j 0 ,j 1 , and j 2 in terms of sin x and cos x.

3. From Problems 13.3 and 13.5, Y 1/2 (x) = − p2/πx cos x. As in Problem 2, obtain Y 3/2 and Y 5/2 and verify the formulas (17.4) for y 0 ,y 1 , and y 2 in terms of sin x and cos x.

4. Using (17.3) and the results stated in Problems 2 and 3 for J 1/2 and Y 1/2 (= N 1/2 ), show that

r I 1/2 2 (x) = sinh x,

5. Show from (17.4) that h (1)

n (x) = −ix

x dx

6. Using (16.1) and (17.4) show that the spherical Bessel functions satisfy the differ- ential equation

x 2 y ′′ + 2xy ′ + [x 2 − n(n + 1)]y = 0. 7. (a)

Solve the differential equation xy ′′ = y using (16.1), and then express the answer in terms of a function I p by (17.3).

(b) As in (a), find a solution of of y ′′

−x 4 y = 0.

8. Using (16.1) and (16.2), verify that (a)

the solution of (17.5) is (17.6); (b)

the solution of (17.8) is (17.9). 9. Using (17.3) and (15.1) to (15.5), find the recursion relations for I p (x). In particular,

show that I ′ 0 =I 1 .

10. Computer plot (a)

I 0 (x), I 1 (x), I 2 (x), from x = 0 to 2.

(b)

K 0 (x), K 1 (x), K 2 (x), from x = 0.1 to 2.

(c) Ai(x) from x = −10 to 10.

598 Series Solutions of Differential Equations Chapter 12

(d) Bi(x) from x = −10 to 1.

11. From (17.4), show that h (1) 0 (ix) = −e −x /x.

Use the Section 15 recursion relations and (17.4) to obtain the following recursion relations for spherical Bessel functions. We have written them for j n , but they are valid for y n and for the h n ’s.

12. j n−1 (x) + j n+1 (x) = (2n + 1)j n (x)/x 13. (d/dx)j n (x) = nj n (x)/x − j n+1 (x) 14. (d/dx)j n (x) = j n−1 (x) − (n + 1)j n (x)/x

15. (d/dx)[x n+1 j n (x)] = x n+1 j n−1 (x) 16. (d/dx)[x −n j n (x)] = −x −n j n+1 (x)