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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

share|improve this question
    
Have you looked here? books.google.com/books?id=Mlk3FrNoEVoC –  Steve Huntsman Mar 24 '10 at 18:56
    
Watson's masterpiece is available online archive.org/details/treatiseontheory00watsuoft –  Jacques Carette Mar 24 '10 at 23:30

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