Complex show differentiable only at z=0

[saraaaahhhhhh]

Homework Statement

Show that f(z) = zRez is differentiable only at z=0,
find f'(0)

The Attempt at a Solution

This should be easy. I find the limit as z_0 approaches 0 of [f(z+z_0) - f(z)]/(z_0) for this function...expand it out, simplify, and find what the limit is when z_0 is purely imaginary vs purely real. I did this, and I got x for the real part and x_0 + 2x + iy for the imaginary part. They're different, so it's not differentiable everywhere.

But apparently it's differentiable only at z=0. So I tried plugging in z=0 and resolving, and I got different values for z_0 real and imaginary.

I get, for the limit when z_0 is only real: 1
When z_0 only imaginary: x_0.

What's going on? Is there some other way to differentiate a complex function that's not using the limit?

 

[gabbagabbahey]

This should be easy. I find the limit as z_0 approaches 0 of [f(z+z_0) - f(z)]/(z_0) for this function...expand it out, simplify, and find what the limit is when z_0 is purely imaginary vs purely real. I did this, and I got x for the real part and x_0 + 2x + iy for the imaginary part. They're different, so it's not differentiable everywhere.

Try your expansion and simplification again. When $z_0$ is real you should find that $$\lim_{z_0\to 0} \frac{f(z+z_0) - f(z)}{z_0}=2x+iy$$ and when $z_0$ is imaginary you should find that $$\lim_{z_0\to 0} \frac{f(z+z_0) - f(z)}{z_0}=x$$

 

[saraaaahhhhhh]

I mixed up my original message. I actually got x for the imaginary part and x_0 + 2x + iy for the real part. I still don't see how the x_0 was eliminated in your version of the expansion, above.

But the main issue is the fact that I get different values in the second part, evaluating the limit at z=0. I get different values for the real and imaginary parts, which means the limit doesn't exist. But according to the problem it should exist! I must just be doing the math wrong.

Am I at least on the right track with the method of finding the limit as z_0 goes to 0, setting z=0, and solving it all out and seeing if the real and imaginary parts of z_0 both go to the same value?

Thanks again for the help!

 

[gabbagabbahey]

I still don't see how the x_0 was eliminated in your version of the expansion, above.

Doesn't x_0 go to zero when z_0 goes to zero?

But the main issue is the fact that I get different values in the second part, evaluating the limit at z=0. I get different values for the real and imaginary parts, which means the limit doesn't exist. But according to the problem it should exist! I must just be doing the math wrong.

You are doing the math wrong!

At z=0, x and y are both zero too aren't they?...Last time I checked 0=0 was a true statement