Cauchy-Riemann and Analytic Functions

Question

Using the Cauchy-Riemann conditions, tell if $f(z) = z^{*}$ is analytic

I have tried this:

$Z = x + iy$

$f(x + iy) = Z^{*} = x - iy$

$U(x,y) = x$

$V(x,y) = -y$

$U_x = 1$ Deriving respect to $x$

$V_x = 0$ Deriving respect to $x$

$U_y = 0$ Deriving respect to $y$

$V_y = -1$ Deriving respect to $y$

Then $U_x <> V_y$ and I suppose that $f(z) = z^{*}$ is not analytic. But I'm not sure, I think I'm wrong. Could you help me, please?

Answer 1

We have that $V_y = -1$. Since $U_x \not = V_y$, the Cauchy Riemann equations are not satisfied.