BESSEL’S EQUATION

12. BESSEL’S EQUATION

Like Legendre’s equation, Bessel’s equation is another of the “named” equations which have been studied extensively. There are whole books on Bessel functions, and you will find numerous formulas, graphs, and numerical values available in your computer program and in reference tables. You can think of Bessel functions as being something like damped sines and cosines. In fact, if you had first learned

about sin nx and cos nx as power series solutions of y ′′ = −n 2 y instead of in ele- mentary trigonometry, you would not feel that Bessel functions were appreciably more difficult or strange than trigonometric functions. Like sines and cosines, Bessel functions are solutions of a differential equation; they can be represented by power series, their graphs can be drawn, and many formulas involving them (compare trigonometric identities) are known. Of special interest to science students is the fact that they occur in many applications. The following list of some of the prob- lems in which they arise will give you an idea of the great range of topics which may involve Bessel functions: problems in electricity, heat, hydrodynamics, elas- ticity, wave motion, quantum mechanics, etc., involving cylindrical symmetry (for this reason Bessel functions are sometimes called cylinder functions); the motion of a pendulum whose length increases steadily; the small oscillations of a flexible chain; railway transition curves; the stability of a vertical wire or beam; Fresnel integrals in optics; the current distribution in a conductor; Fourier series for the arc of a circle. We shall discuss some of these applications later (see Section 18, and Chapter 13, Sections 5 and 6).

588 Series Solutions of Differential Equations Chapter 12

Bessel’s equation in the usual standard form is

x 2 y ′′ + xy ′ + (x 2 −p 2 )y = 0,

where p is a constant (not necessarily an integer) called the order of the Bessel func- tion y which is the solution of (12.1). You can easily verify that x(xy ′ ) ′ =x 2 y ′′ +xy ′ , so we can write (12.1) in the simpler form

x(xy ′ ) ′ + (x 2 −p 2 )y = 0.

We find a generalized power series for (12.2) in the same way that we solved (11.2). [In fact, (11.2) is a form of Bessel’s equation! See Problems 16.1 and 17.1.] Writing only the general terms in the series for y and the derivatives we need in (12.2), we have

a n (n + s)x n+s−1

a n (n + s) 2 x n+s−1

n=0 ∞

x(xy ′ ) ′ =

a n (n + s) 2 x n+s

n=0

We substitute (12.3) into (12.2) and tabulate the coefficients of powers of x:

x s+n x(xy ′ ) ′

The coefficient of x s gives the indicial equation and the values of s:

s 2 −p 2 = 0,

s = ±p.

The coefficient of x s+1 gives a 1 = 0. The coefficient of x s+2 gives a 2 in terms of a 0 , etc., but we may as well write the general formula from the last column at this point. We get

[(n + s) 2 −p 2 ]a n +a n−2 =0

or

a n n−2 =−

(n + s) 2 −p 2

Section 12 Bessel’s Equation 589

First we shall find the coefficients for the case s = p. From (12.4) we have

a a (12.5)

n(n + 2p) Since a 1 = 0, all odd a’s are zero. For even a’s it is convenient to replace n by 2n;

(n + p) 2 −p 2

n + 2np

then from (12.5) we have

2n =−

a 2n−2

2n−2

2n(2n + 2p)

2 2 n(n + p)

The formulas for the coefficients can be simplified by the use of the Γ function notation (Chapter 11, Sections 2 to 5) as you can see by examining (12.7) below. Recall that Γ(p + 1) = pΓ(p) for any p, so,

Γ(p + 2) = (p + 1)Γ(p + 1), Γ(p + 3) = (p + 2)Γ(p + 2) = (p + 2)(p + 1)Γ(p + 1),

and so on. Then from (12.6) we find

(1 + p)(2 + p)

a 6 =−

3!2(3 + p) 3!2 6 (1 + p)(2 + p)(3 + p)

and so on. Then the series solution (for the s = p case) is

2!Γ(3 + p) 2 3!Γ(4 + p) 2

(12.8) p x p

=a 0 2 Γ(1 + p)

2 Γ(1)Γ(1 + p) − Γ(2)Γ(2 + p) 2

Γ(3)Γ(3 + p) 2 − Γ(4)Γ(4 + p) 2 +··· . We have inserted Γ(1) and Γ(2) (which are both equal to 1) in the first two terms

and written x p =2 p (x/2) p to make the series appear more systematic. If we take

then y is called the Bessel function of the first kind of order p, and written J p (x).

590 Series Solutions of Differential Equations Chapter 12

1 x 2+p J p (x) =

Γ(1)Γ(1 + p) 2 Γ(2)Γ(2 + p) 2

1 x 6+p (12.9)

1 x 4+p

+··· Γ(3)Γ(3 + p) 2 Γ(4)Γ(4 + p) 2

(−1) n

x 2n+p

n=0 Γ(n + 1)Γ(n + 1 + p) 2

PROBLEMS, SECTION 12

1. Show by the ratio test that the infinite series (12.9) for J p (x) converges for all x. Use (12.9) to show that:

2. J 2 (x) = (2/x)J 1 (x) − J 0 (x)

3. J 1 (x) + J 3 (x) = (4/x)J 2 (x)

4. (d/dx)J 0 (x) = −J 1 (x)

5. (d/dx)[xJ 1 (x)] = xJ 0 (x)

6. J 0 (x) − J 2 (x) = 2(d/dx)J 1 (x)

7. lim J

1 (x)/x = 1 2

x→0

8. x→0 lim x −3/2 J 3/2 (x) = 3 −1 p2/π Hint: See Chapter 11, equations (3.4) and (5.3). 9. pπx/2 J 1/2 (x) = sin x