Complex problem f(z) = sqrt(|xy|) in x + iy form?

[saraaaahhhhhh]

Homework Statement

In the title: f(z) = sqrt(|xy|)...show that this satisfies the Cauchy-Riemann equations at z=0, but is not differentiable there.

Homework Equations

Cauchy-Riemann just states that partial u partial x = partial v partial y and partial u partial y = - partial v partial x.

The Attempt at a Solution

I think all the partials du/dx, du/dy, dv/dx, and dv/dy are 0. Because f(0) is 0 in this case, right? Maybe that's wrong. Maybe I'm assumign too much by saying that if z=0, x and y also are 0.

Anyway, then I just need to show the limit as h->0 for differentiation doesn't exist, right? If what I said above is right, than it would just be plugging in some x_0 and y_0 (if h = x_0 + iy_0) for x and y and than finding the limit for when x_0 = 0 and when y_0 = 0, and as long as those aren't equal then it's not differentiable.

Does this sound right, or totally off?

 

[gabbagabbahey]

I think all the partials du/dx, du/dy, dv/dx, and dv/dy are 0. Because f(0) is 0 in this case, right? Maybe that's wrong.

You're supposed to take the partial derivatives first and then evaluate them at z=0; not the other way around!

 

[saraaaahhhhhh]

Wouldn't that mean the Cauchy-Riemann equations don't hold? I'm a little unsure on what u would be in this case.

Do I need to separate sqrt(|xy|) into the real and imaginary parts? Or can I just assume all is real and then take the partial derivates of u, and the partials of v would just be 0, since there's no complex part?

This is why I think Cauchy-Riemann wouldn't hold, but I'm not sure.

Thanks so much for the response! I do appreciate it.

 

[saraaaahhhhhh]

Just to go ahead and try this: would partial u partial x be y(xy)^(-1/2)?
And then partial u partial y be x(xy)^(-1/2)?
And both partials of v be 0? This is assuming that sqrt(xy) is just the 'real' part...if f(z) takes the form u + iv.

I have a feeling this is wrong, since Cauchy-Riemann is supposed to hold and doesn't hold using this method. But if it's wrong, how am I supposed to figure out what u and v are in cases like this?

 

[gabbagabbahey]

Do I need to separate sqrt(|xy|) into the real and imaginary parts? Or can I just assume all is real and then take the partial derivates of u, and the partials of v would just be 0, since there's no complex part?

Yes, x and y are real numbers, so |xy| is a positive, real number and so sqrt(|xy|) is real.

This is why I think Cauchy-Riemann wouldn't hold, but I'm not sure.

Careful, what do you get for $$\frac{\partial u}{\partial x}$$ and $$\frac{\partial u}{\partial y}$$...where do they equal zero?

 

[saraaaahhhhhh]

I'm not sure what you're saying here. I think you get that they don't exist or are undefined, because there would be 0 on the bottom of a fraction? But I'm not sure where you're going with this.