Let the complex-valued function $f(z)$ be analytic in the neighbourhood of the point $z_{0}$ and let $u(x, y)$ be the real part of $f(z)$. Show that

$$f(z)=2 u\left(\frac{z+\bar{z}_{0}}{2}, \frac{z-\bar{z}_{0}}{2 i}\right)-\overline{f\left(z_{0}\right)}, \quad z=x+i y$$

Hence find the analytic function whose real part is

$$e^{-y}[x \cos x-y \sin x]$$