Page 316 - Schaum's Outline of Differential Equations

CHAP. 30] GAMMA AND BESSEL FUNCTIONS 299

Thus,
and, in general,

2
2
The indicial equation is X - p = 0, which has the roots 'k\=p and X 2 = —p (p nonnegative).
Substituting 'k = p into (_/) and (2) and simplifying, we find that a l = 0 and

Hence, 0 = aj = a 3 = a s = a 7 = • • • and

and, in general,

Thus,

It is customary to choose the arbitrary constant a 0 as a 0 = . Then bringing a Qx p inside the brackets
and summation in (3), combining, and finally using Problem 30.4, we obtain

30.10. Find the general solution to Bessel's equation of order zero.
2
2
For p = 0, the equation is x y" + xy' + x y = 0, which was solved in Chapter 28. By (4) of Problem 28.10, one solution is

Changing n to k, using Problem 30.6, and letting = 1 as indicated in Problem 30.9, it follows that
y\(x) = JQ(X). A second solution is [see (_/) of Problem 28.11, with a 0 again chosen to be 1]