cv.complex variables - Bessel function in polar coordinates

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

$J_alpha(z)=|J_alpha(z)|e^{ivarphi_alpha(z)}$

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