I have the following functions: $f:\mathbb{C} \to \mathbb{C}$ and $g:\mathbb{C} \to \mathbb{C}$, defined as
$$f(z)=z^2e^{\bar{z}}, \quad g(z)=\sin(z)f(z)$$
I am trying to solve this using Cauchy-Riemann equations, but I am struggling with the definition of $u$ and $v$
$$ u(x,y)= \text{Re}[f(x+iy)] \;\;\;\; v(x,)= \text{Im} [f(x+iy)] $$
Any hints? Should I try a different approach?
That's not the best approach.
It should be clear that $f$ is differentiable at $0$. If $z\ne0$; then $f$ is differentiable at $z$ if and only if $\varphi(z)=e^{\overline z}$ is differentiabl at $z$. And it is much easier to apply the Cauchy-Riemann equations method to $\varphi$. If $x,y\in\Bbb R$, then$$\varphi(x+yi)=e^x(\cos(y)-\sin(y)i)$$and therefore you should take $u(x,y)=e^x\cos(y)$ and $v(x,y)=-e^x\sin(y)$.