If we have some $h:\mathbb{C}\rightarrow \mathbb{C},$ such that h is defined to be $h(z)= x^3y^2+ix^2y^3, \text{where} \space z=x+iy,$ find all set of complex numbers where $h $ is differentiable.
So what I have done so far was to check using Cauchy - Riemann equations, and for the second part of the Cauchy - Riemann, when I checked $u_y=-v_x,$ this holds iff $x=0$, or $y =0.$ Then does this mean that it works for all complex numbers having $x=0, \text{or}\space y = 0? $ I would appreciate the help, with this problem.
Yes. Since the functions $u_x$, $u_y$, $v_x$, and $v_y$ are continuous, $h$ is differentiable at those points which are solution of the Cauchy-Riemann equations and only at those points. So, $h'(z)$ exists if and only if $\operatorname{Re}z=0$ or $\operatorname{Im}z=0$.