I want to write the Bessel function of the first kind in polar coordinates

$J_\alpha(z)=|J_\alpha(z)|e^{i\varphi_\alpha(z)}$

Is anything known about $\varphi_\alpha(z)$?

In particular, I'm interested in proving that

$\varphi_\alpha(z)\neq \varphi_{\alpha+1}(z)$

for all non-real $x$, where equalty is taken modulo $\pi$.

Thank you,

Afonso