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

• saraaaahhhhhh
In summary, the conversation discusses the Cauchy-Riemann equations and the differentiability of the function f(z) = sqrt(|xy|) at z=0. It is determined that the function satisfies the Cauchy-Riemann equations at z=0, but is not differentiable there. Various approaches are discussed, including evaluating the partial derivatives at z=0 and taking the limit as h->0, and separating sqrt(|xy|) into its real and imaginary parts. It is also pointed out that x and y are real numbers, so sqrt(|xy|) is also real.
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?

saraaaahhhhhh said:
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!

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.

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?

saraaaahhhhhh said:
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?

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.

## 1. What is a complex problem?

A complex problem is a mathematical problem that involves complex numbers, which are numbers that have both a real and imaginary component. These problems often require knowledge of algebra, geometry, and calculus to solve.

## 2. What is the function f(z) = sqrt(|xy|) in x + iy form?

The function f(z) = sqrt(|xy|) in x + iy form is a complex function that takes in a complex number, z, and returns the square root of the absolute value of the product of the real and imaginary parts of z. It is written in the form of x + iy, where x is the real part and iy is the imaginary part.

## 3. How do you solve a complex problem using the function f(z) = sqrt(|xy|) in x + iy form?

To solve a complex problem using the function f(z) = sqrt(|xy|) in x + iy form, you would substitute the given complex number, z, into the equation and simplify. This would result in a new complex number, which would be the solution to the problem.

## 4. What are some real-life applications of the function f(z) = sqrt(|xy|) in x + iy form?

The function f(z) = sqrt(|xy|) in x + iy form has several real-life applications, including in electrical engineering, physics, and signal processing. It can also be used to model wave propagation and calculate the electric and magnetic fields in electromagnetic radiation.

## 5. What are some strategies for solving complex problems involving the function f(z) = sqrt(|xy|) in x + iy form?

Some strategies for solving complex problems involving the function f(z) = sqrt(|xy|) in x + iy form include breaking the problem down into smaller parts, using properties of complex numbers to simplify the equation, and drawing diagrams or graphs to visualize the problem. It is also helpful to have a good understanding of algebra, geometry, and calculus concepts and techniques.

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