I have a situation where I have the product of two modified Bessel functions with complex arguments, as given below:
$K_n(a+ib)I_n(x+iy)$
It is required for me to take the real part of this product, which prompts me to look into the Kelvin functions. However, these are only defined as the real and imaginary parts of the Bessel functions for specific complex numbers. I am aware though that the following will still hold for any complex number $z$:
$\text{ber}_n(z)+i\text{bei}_n(z) = e^{n\pi i/2}I_n(ze^{\pi i/4})$
$\text{ker}_n(z)+i\text{kei}_n(z) = e^{-n\pi i/2}K_n(z e^{\pi i/4})$
where the Kelvin functions themselves will also be complex. My Bessel function product above (rewriting the arguments) would then result in:
$I_n(z_1 e^{\pi i/4})K_n(z_2 e^{\pi i/4}) = \text{ber}_n(z_1)\text{ker}_n(z_2) - \text{bei}_n(z_1)\text{kei}_n(z_2)+ i(\text{ber}_n(z_1)\text{kei}_n(z_2)+\text{bei}_n(z_1)\text{ker}_n(z_2))$
How then can I find the real parts of the right hand side if the Kelvin functions are also complex? Thanks in advance!