How to separate wavenumber $k$ into real and imaginary parts?

$k^2$ is easy and I assume you know how to expand that into its real and imaginary parts. As for $ln k^2$, use the fact that (the principal branch of) $ln(r e^{itheta}) = ln(r) + itheta$. So, if you can write $k^2$ in the form $r e^{itheta}$, you're done. Now, writing a complex number $a + i b$ in its polar form $r e^{itheta}$ isn't hard: $r = sqrt{a^2 + b^2}$ and $theta = arctan left( dfrac{b}{a} right)$.