# Thread: Show that f(Z) = Z(hat) is not differentiable.

1. ## Show that f(Z) = Z(hat) is not differentiable.

Hi ppl, would love the solutuin to "show that f(Z) = Z(hat) is not differentiable"

2. Originally Posted by davy337
Hi ppl, would love the solutuin to "show that f(Z) = Z(hat) is not differentiable"

I think you mean $f(z)=\bar{z}=x-iy$ . Use the Cauchy Riemann equations for $u=x,v=-y$ .

3. Originally Posted by davy337
Hi ppl, would love the solutuin to "show that f(Z) = Z(hat) is not differentiable"
The previous poster has given help. If you need more help, please show all your work and say where exactly you are stuck.

4. Or you could use the reasoning that leads to the Cauchy-Riemann equations. Let z= x+ iy where x and y are real numbers. Then $\overline{z}= x- iy$. To find the derivate at, say $z_0= x_0+ iy_0$ we form the "difference quotient" $\frac{f(z)- f(z_0)}{z- z_0}=\frac{x- iy- (x_0- iy_0)}{x+ iy- (x_0+ iy_0)}= \frac{x- x_0+ i(-y- y_0)}{x-x)+ i(y- y_0)}$ and take the limit as z goes $z_0= x_0+ iy_0$. Since that is a two dimensional limit, in order that the derivative exist, the limit must be the same as we approach from any direction.

In particular, what do we get if we approach $(x_0, y_0)$ along the line $(x, y_0)$ ?

What do we get if we approach $(x_0, y_0)$ along the line $(x_0, y)$? Are those the same?

5. Originally Posted by FernandoRevilla
I think you mean $f(z)=\bar{z}=x-iy$ . Use the Cauchy Riemann equations for $u=x,v=-y$ .
There is also a useful (but perhaps not commonly known) Cauchy-Riemann relation for when you have a function of $z$ and $\overline{z}$:

$\displaystyle \frac{\partial f}{\partial \overline{x}} = 0$.