= Solution
The function $h(z)=1/(z+1)$ is <holomorphic> on $U$ and has
$$
\frac1{z+1}=\frac{x+1-iy}{(x+1)^2+y^2}.
$$
Its imaginary part is exactly the prescribed $v$. Thus $f-h$ is a real-valued <holomorphic function>. The <Cauchy-Riemann equations> force both partial derivatives of its real part to vanish. Since $\mathbb C\setminus\{-1\}$ is a <connected space>, $f-h$ is a single real constant. All possibilities are therefore
$$
\boxed{f(z)=\frac1{z+1}+C,\qquad C\in\mathbb R}.
$$
Conversely every such function has the stated imaginary part. The data determine $f$ only up to this real additive constant.
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