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
