I'm trying to write the function $f(z)= (e^z-1)/z$ when $z \neq 0$ and 1 when $z=0$ in the form of $u(x,y)+iv(x,y)$ where $z=x+iy$. How should I proceed?
I have already shown that it is differentiable (continuously) for all $z \in \mathbb C$ in an earlier problem, but is this needed to solve this?
Also, another problem asks to show that $u(x,y)$ is $C^\infty$; this is simply a result of the fact that $(e^z)'=e^z$ correct? Thank you.