Graduate Bessel functions of imaginary order

[KariK]

TL;DR

Is it true that the complex conjugate of Hankel function (H) of first kind of order ia is equal to H of second kind of order -ia, so H of first kind and its complex conjugate are linearly independent.

In Wikipedia article on Bessel functions there is an integral definition of “non-integer order” a (“alpha”). For imaginary order ia I get that Jia* = J-ia, where * is complex conjugate and ia and -ia are subscripts. Then in same article there is a definition of Neumann function, again for “non-integer order”. If I again use it for imaginary order, I get that Yia* = Y-ia. So then for Hankel functions I get that H^1ia* = H^2-ia.

Is that valid for imaginary orders and does it then mean that H^1 and its complex conjugate are linearly independent? If the above is not valid, is there some other way to show the independence?

 

[Paul Colby]

Bessel function Wiki

Yes, they are linearly independent. For integral and half integral $\alpha$ the Hankel are the radial wave functions for incoming and outgoing waves. In the differential equation (first equation in the linked article) $\alpha$ appears as $\alpha^2$. This implies several things about the equations solutions. First, solutions are analytic functions of $\alpha$ and second, both $y_\alpha(x)$ and $y_{-\alpha}(x)$ obey the very same differential equation. Being a linear second order differential, the are at most two linearly independent solutions, all others expressible as a linear combination.

 

[KariK]

Thank you jedishrfu for moving this thread to a more appropriate place. While I got the Bessel equation from physics, the topic is indeed more math than physics. And thank you Paul Colby for confirming my assumptions.