# Bessel functions of imaginary order

• A
• KariK

#### KariK

TL;DR Summary
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?

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.

• topsquark
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.

• jedishrfu