Page 318 - Schaum's Outline of Differential Equations

CHAP. 30] GAMMA AND BESSEL FUNCTIONS 301

30.13. Prove That
We may differentiate the series for the Bessel function term by term. Thus,

Noting that 2T(k + p + 2) = 2(k + p + l)T(k + p + 1) and that the factor 2(k + p + 1) cancels, we have

For the particular case p = 0, it follows that

30.14. Prove thatxJ p(x) = pJ p(x)-xJ p + l(x).
We have

Using Problem 30.12 on the last summation, we find

For the particular case p = 0, it follows that xJ$(x) = -xJi(x), or

30.15. Prove that xJ' p(x) = -pJ p(x) + xJ^x).