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- 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?

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?