Letting $J_\nu$ denote the Bessel function of order $\nu$, if $0 < a < b$ will the first zero of $J_{0}$ be less than the first zero of $J_a$ be less than the first zero of $J_b$? Does this hold for the $k$th zero in general?
I am solving the radial Schrodinger equation with potential $\exp(-r/a)$ and encountered a quantization condition involving the zeroes of the Bessel function. I have looked over some tables for the zeroes of Bessel functions and see that they generally follow that pattern, so I am curious if there is a way to prove/disprove that statement, especially for non-integer $\nu$.