Is There a Complex Gradient Formula for Polar Coordinates?

[gaganaut]

Does there exist anything like a polar complex differentiation? So there exists a gradient equation in polar coordinates something like
$$\nabla{f} = \frac{\partial f}{\partial r} e_r + \frac{1}{r}\;\frac{\partial f}{\partial \theta} e_{\theta}$$

But this is not for a complex number $$f(z)$$ where $$z=r\,e^{i\theta}$$. Now for cartesian coordinates, there exists a complex gradient formula as
$$\nabla{f}(z) = \frac{\partial f}{\partial x} e_x - i\;\frac{\partial f}{\partial y} e_y$$

So I would like to know if there exists a formula like $$\nabla{f}(z) = \frac{\partial f}{\partial r} e_r -i\; \frac{1}{r}\;\frac{\partial f}{\partial \theta} e_{\theta}$$, if $$z=r\,e^{i\theta}$$.

I can differentiate by $$z$$ directly. But I would like to know if anything like this exists.

Thanks

 

[fresh_42]

Here's a brief paper on Cauchy Riemann equation in polar coordinates.
https://users.math.msu.edu/users/shapiro/Teaching/classes/425/crpolar.pdfand here the gradient:
http://math.mit.edu/classes/18.013A/HTML/chapter09/section04.html