# Complex operator in polar coords

1. Sep 1, 2011

### demidemi

1. The problem statement, all variables and given/known data
If z=x + iy, what is d/dz in polar coordinates?

3. The attempt at a solution

I know that expanded,

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

Where to go from there?

2. Sep 2, 2011

### HallsofIvy

Staff Emeritus
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 $\theta= arctan(y/x)$.

3. Sep 4, 2011

### demidemi

Why is "f" necessary there?

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

Last edited: Sep 4, 2011