The modified Bessel differential equation is:
$$x^2y" + xy' - (x^2 + n^2)y = 0$$
I want to solve a differential equation and then calculate the Fourier transform of slightly different equation: $$x^2y" + xy' - (cx^2 - n^2)y = 0$$
I found that the solution to the above equations can be written in terms of solutions of the first equation in the following way:
$$K_{in} = \frac{\pi}{2 \sinh{\pi n}} [I_{-in}(x) - I_{in}(x)]$$
I wanted to calculate the Fourier transform of $K_{ia}$ but since the order of the Modified Bessel function is imaginary, I am not sure what to do. Can anyone point out any reference?