We know that rectangular co-ordinates $(x, y)$ can be written as a complex number $re^{i\theta}$ where
$r = \sqrt{x^2 + y^2}$
and
$\theta = \tan^{-1} \big(\frac{y}{x}\big)$
and $r,\theta \in \mathbb{R}$ represent polar co-ordinates of the same point. Does it make sense to define 'complex polar co-ordinates' by making $r$ and $\theta$ complex numbers? Let's define
$r = r_0 + i r_1$
and
$\theta = \theta_0 + i \theta_1$
then after a lot of manipulation, we get:
$r e^{i \theta} = e^{-\theta_1} [ r_0 \cos \theta_0 - r_1 \sin \theta_0 + i ( r_0 \sin \theta_0 + r_1 \cos \theta_0)]$ (i)
If the above makes sense, then since the graph of
$z = f(\theta) e^{i \theta}$
where $r$ and $\theta$ are real and
$f(\theta) = r$
is a circle, the graph of
$z = ir e^{i \theta}$
where $r$ and $\theta$ are still reals should be a circle because expansion (i) above reduces to
$r e^{i (\theta + \pi/2)}$
where we have dropped $r_0$ and $\theta_1$ since they are 0 and simplified the notation. Just want to know if the above analysis is correct or not. I tried going over the results but something doesn't feel right...