Let $f$ be analytic in $\{z\in \mathbb{C} | \Re(z)>1\}$ such that $f=u+iv$ and $\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$ in $U$. Show that there exist $c \in \mathbb{R}, d /in \mathbb{C}$ such that $f(z)= -icz +d$.
So if $f$ is analytic it satisfies Cauchy Riemann equations and there fore the partials equality would imply that either $u$ or $v$ are constant. I asked my teacher and he said I should integrate but I'm lost, anything I'm missing?
Thanks