Oh, I've forgot, what is the complex potenial. So,
$w = \varphi + i \psi$
where $\varphi$ is a potential and $\psi$ is a stream function.
Thus, $\boldsymbol v = \text{grad} \varphi \;$:
$v_r = \frac{\partial \varphi}{\partial r} = \text{Re} \left[ \frac{\partial \, w(r e^{i \varphi})}{\partial r} \right] $
$v_{\theta} = \frac{1}{r} \frac{\partial \varphi}{\partial \theta} = \text{Re} \left[ \frac{1}{r} \frac{\partial \, w(r e^{i \varphi})}{\partial \theta} \right] $
You could use stream function $\psi$ instead of potential though.
I've used FriCAS to evaluate things to the answer:
(15) -> D(real(-G*%i/(2*%pi)*log(r*exp(%i*phi))),r) (15) 0 (16) -> D(1/r * real(-G*%i/(2*%pi)*log(r*exp(%i*phi))),phi) G (16) ────── 2%pi r