Page 317 - Schaum's Outline of Differential Equations

300 GAMMA AND BESSEL FUNCTIONS [CHAP. 30

which is usually designated by N 0(x). Thus, the general solution to Bessel's equation of order zero is
y = c^O) + c 2N 0(x).
Another common form of the general solution is obtained when the second linearly independent solution is not
taken to be N 0(x), but a combination of N 0(x) and JQ(X). In particular, if we define

where y is the Euler constant defined by

then the general solution to Bessel's equation of order zero can be given as y = CiJ 0(x) + c 2Y 0(x).

30.11. Prove that

Writing the k = 0 term separately, we have

which, under the change of variables j = k— 1, becomes

The desired result follows by changing the dummy variable in the last summation fromy to k.

30.12. Prove that

Make the change of variables j = k + 1:

Now, multiply the numerator and denominator in the last summation by 2j, noting that j(j — 1)! =j\ and
2 +
22j+ P-i(2) = 2 J P. The result is

Owing to the factory in the numerator, the last infinite series is not altered if the lower limit in the sum is changed from
j = 1 toy = 0. Once this is done, the desired result is achieved by simply changing the dummy index fromy to k.