Let $f(z)=u(x, y)+i v(x, y)$, where $z=x+i y$, be an analytic function of $z$ in a domain $D$ of the complex plane. Derive the Cauchy-Riemann equations relating the partial derivatives of $u$ and $v$.

For $u=e^{-x} \cos y$, find $v$ and hence $f(z)$.