Chapter 12.15, Problem 8P

From equation (15.4), show that ${\displaystyle {\int }_0^{\infty }{J}_1}(x)dx={\displaystyle {\int }_0^{\infty }{J}_3}(x)dx=\cdots ={\displaystyle {\int }_0^{\infty }{J}_{2n+1}}(x)dx,$

And ${\displaystyle {\int }_0^{\infty }{J}_0}(x)dx={\displaystyle {\int }_0^{\infty }{J}_2}(x)dx=\cdots ={\displaystyle {\int }_0^{\infty }{J}_{2n}}(x)dx.$

Then, by Problem 7, show that ${\displaystyle {\int }_0^{\infty }{J}_n}(x)dx=1$ for all integral n.

(15.4) $J_{p-1}(x)-J_{p+1}(x)=2J_p^{\prime }(x),$