Let $F(z)=u(x, y)+i v(x, y)$ where $z=x+i y$. Suppose $F(z)$ is an analytic function of $z$ in a domain $\mathcal{D}$ of the complex plane.

Derive the Cauchy-Riemann equations satisfied by $u$ and $v$.

For $u=\frac{x}{x^{2}+y^{2}}$ find a suitable function $v$ and domain $\mathcal{D}$ such that $F=u+i v$ is analytic in $\mathcal{D}$.