Chapter 12.18, Problem 3P

Prove $\begin{array}{l}\hfill \\ J_p(x)J_{-p}^{\prime }(x)-J_{-p}(x)J_p^{\prime }(x)=-\frac{2}{\pi x}\sin p\pi \hfill \end{array}$ as follows: Write Bessel’s equation (12.1) with $y=J_p$ and with $y=J_{-p}$; multiply the $J_p$ equation by $J_{-p}$ and the $J_{-p}$ equation by $J_p$ and subtract to get .

Then $J_p{J}_{-p}^{\prime }-J_{-p}{J}_p^{\prime }=c/x.$ To find c, use equation (12.9) for each of the four function and pick out the 1/x terms in the products. Then use (5.4) of Chapter 11.