Let a real-valued function $u=u(x, y)$ be the real part of a complex-valued function $f=f(z)$ which is analytic in the neighbourhood of a point $z=0$, where $z=x+i y .$ Derive a formula for $f$ in terms of $u$ and use it to find an analytic function $f$ whose real part is

$$\frac{x^{3}+x^{2}-y^{2}+x y^{2}}{(x+1)^{2}+y^{2}}$$

and such that $f(0)=0$.