Complex operator in polar coords

[demidemi]

Homework Statement

If z=x + iy, what is d/dz in polar coordinates?

The Attempt at a Solution

I know that expanded,

d/dz = 1/2 (d/dx) - i (d/dy)

Where to go from there?

 

[HallsofIvy]

Use the chain rule:

You have
$$\frac{df}{dz}= \frac{\partial f}{\partial x}+ \frac{\partial f}{\partial y}i$$
$$= \left(\frac{\partial f}{\partial r}\frac{\partial r}{\partial x}+ \frac{\partial f}{\partial \theta}\right)+ \left(\frac{\partial f}{\partial r}\frac{\partial r}{\partial y}+ \frac{\partial f}{\partial \theta}\frac{\partial \theta}{\partial y}\right)i$$

with, of course, $r= \sqrt{x^2+ y^2}$ and $\theta= arctan(y/x)$.

 

[demidemi]

Why is "f" necessary there?

Also, should there be a "partial theta partial x" in the first parentheses?