A question about Bessel function

[zetafunction]

if J_{u}(x) is a Bessel function..

do the following functions has special names ?

a) J_{ia}(ib) here 'a' and 'b' are real numbers

b) J_{ia}(x) the index is complex but 'x' is real

c) J_{a}(ix) here 'x' is a real number but the argument of the Bessel function is complex.

 

[Mute]

Bessel functions $J_\nu(z),~Y_\nu(z)$ of real order and imaginary argument are related to the modified Bessel functions $I_\nu(z),~K_\nu(z)$ in a similar way as sine and cosine are related to sinh and cosh.

For imaginary order, see this Bessel function subpage on the Digital Library of Mathematical functions; for imaginary order and imaginary argument (i.e., the modified Bessel functions of imaginary order), see this page.