Exponential of complex variable

Question

What is the equivalent to " $(e i)^z$ " , where i is the imaginary "i" and z is a variable (maybe a complex one) ? (I'm thinking in a possible symmetry with $e^{iz}$)

Answer 1

Here is how you advance

$$(ie)^z = e^{z\ln(ei)}=e^{z(\ln(e)+i(\frac{\pi}{2}+2k\pi i) } = e^{z(1 + \frac{i\pi}{2} - 2k\pi ) } = \dots,\quad k \in \mathbb{Z}. $$