Finding the $z$ in $\mathbb{C}$ where the Cauchy Riemann equations hold for $f.$

Question

I'm new to complex analysis and struggling with the application of Cauchy Riemann equation. I've been given the following question.

Given $z=x+iy$ and $f(z) = x^3 + 3xy^2 + i(y^3 + 3x^2y)$

For which $z$ in $\mathbb{C}$ are the Cauchy Riemann equations for this $f$ satisfied?

I'm not sure if I understand the question correctly.

How I've gone about this so far is taking

$u(x,y) = x^3 + 3xy^2$

$v(x,y) = y^3 + 3x^2y$

and using $u_y = -v_x $ to get

$6xy = -6xy$

$\implies z$ satisfies Cauchy Riemann equations if $x = 0 \lor y = 0$

Is this the correct way of tackling this question?

(Sidenote: Currently watching this playlist to get a better understanding but speeding through the vids as I'd like to get to complex integration by Monday. Any other recommended series on Complex Analysis would be appreciated)

Answer 1

Well, there is also the equation $u_x=v_y$, but this one is satisfied everywhere.

So, yes, the complex numbers $z$ such that the Cauchy-Riemann equations are satisfied at $z$ are those such that $\operatorname{Re}(z)=0$ or that $\operatorname{Im}(z)=0$.