THE SECOND SOLUTION OF BESSEL’S EQUATION

13. THE SECOND SOLUTION OF BESSEL’S EQUATION

We have found just one of the two solutions of Bessel’s equation, that is, the one when s = p; we must next find the solution when s = −p. It is unnecessary to go through the details again; we can just replace p by −p in (12.9). In fact, the

solution when s = −p is usually written J −p . From (12.9) we have

x 2n−p (13.1)

(−1) n

J −p (x) =

n=0 Γ(n + 1)Γ(n − p + 1) 2

If p is not an integer, J p (x) is a series starting with x p and J −p is a series starting with x −p . Then J p (x) and J −p (x) are two independent solutions and a linear combination of them is a general solution. But if p is an integer, then the first few terms in J −p are zero because Γ(n − p + 1) in the denominator is Γ of a negative integer, which is infinite. You can show (Problem 2) that J −p (x) starts with the term x p (for integral p) just as J p (x) does, and that

for integral p; thus J −p (x) is not an independent solution when p is an integer. The second solution

−p p (x) = (−1) J p (x)

in this case is not a Frobenius series (11.1) but contains a logarithm. J p (x) is finite at the origin, but the second solution is infinite and so is useful only in applications involving regions not containing the origin.

Although J −p (x) is a satisfactory second solution when p is not an integer, it is customary to use a linear combination of J p (x) and J −p (x) as the second solution.

Section 14 Graphs and Zeros of Bessel Functions 591

This is much as if sin x and (2 sin x − 3 cos x) were used as the two solutions of y ′′ + y = 0 instead of sin x and cos x. Remember that the general solution of

this differential equation is a linear combination of sin x and cos x with arbitrary coefficients. But A sin x + B(2 sin x − 3 cos x) is just as good a linear combination as

c 1 sin x + c 2 cos x. Similarly, any combination of J p (x) and J −p (x) is a satisfactory second solution of Bessel’s equation. The combination which is used is called either the Neumann or the Weber function and is denoted by either N p or Y p :

cos(πp)J p (x) − J (x)

it has a limit (as p tends to an integral value) which gives a second solution. This is why the special form (13.3) is used; it is valid for any p. N p or Y p are called Bessel functions of the second kind. The general solution of Bessel’s equation (12.1) or (12.2) may then be written as

y = AJ p (x) + BN p (x),

where A and B are arbitrary constants.

PROBLEMS, SECTION 13

1. Using equations (12.9) and (13.1), write out the first few terms of J 0 (x), J 1 (x), J −1 (x), J 2 (x), J −2 (x). Show that J −1 (x) = −J 1 (x) and J −2 (x) = J 2 (x).

2. Show that, in general for integral n, J −n (x) = (−1) n J n (x), and J n (−x) = (−1) n J n (x). Use equations (12.9) and (13.1) to show that: 3. pπx/2 J −1/2 (x) = cos x 4. J 3/2 (x) = x −1 J 1/2 (x) − J −1/2 (x). 5. Using equation (13.3), show that N 1/2 (x) = −J −1/2 (x); that N 3/2 (x) = J −3/2 (x). 6. Show from (13.3) that N

(2n+1)/2 (x) = (−1)

n+1

J −(2n+1)/2 (x).