Chapter 12.19, Problem 5P

We obtained (19.10) for $J_p(x),p\ge 0.$ It is, however, valid for $p>-1$, that is for $N_p(x),0\le p<1.$ The difficulty in the proof occurs just after (19.7); we said that $u,v,u^{\prime },v^{\prime }$ are finite at $x=0$ which is not true for $N_p(x).$ However, the negative powers of $x$ cancel if $p<1.$ Show this for $p=\frac{1}{2}$ by using two terms of the power series (12.9) or (13.1) for the function $N_{1/2}(x)=-J_{-1/2}(x)$ [see (13.3)].