Let $$\Omega = \{z \in \mathbb{C} : \textrm{Im}(z) > 0\} \setminus \{iy : y \geq 1\}.$$ I need to find a bounded harmonic function $u \colon \Omega \to \mathbb{R}$ such that for each $x \in \mathbb{R}$ we have $u(x + iy) \to 0$ and $u(z) \to 1$ as $\inf\{\vert z - iy \vert : y \geq 1\} \to 0$. I thought the thing to do was take the imaginary part of some branch of $\log$. In order to have $u(x + iy) \to 0$ for each $x$, I need to map the real axis to a single ray, so I thought first to apply $z \to z^2$ to map $\Omega$ to $\mathbb{C} \setminus ((-\infty,-1] \cup [0,\infty))$. But this won't work because the logarithm is discontinuous at the branch cut.
I'm not sure what to do. Normally given a problem where you need to construct a harmonic function which takes fixed values on two lines, I would try to use a Mobius transformation to map the lines to concentric circles or parallel lines and then apply $\log \vert z \vert$ or take real/imaginary parts. But that doesn't work here since the lines in question intersect.