Chapter 12.19, Problem 4P

Define $J_p(z)$ for complex $z$ by the power series (12.9) with $x$ replaced by $z.$ (By Problem 12.1, the series converges for all z.) Show by ( 19.10 ) that all the zeros of $J_p(z)$ are real. Hint: Suppose $\alpha$ and $\beta$ in (19.10) were a complex conjugate pair; show that then the integrand would be positive so the integral could not be zero.