= Solution
A <point vortex> of circulation $\Gamma$ at $z_0$ contributes the <complex potential of a two-dimensional potential flow> $-i\Gamma\log(z-z_0)/(2\pi)$ with the stated convention. The <method of images> places a vortex of the opposite circulation at $\overline z_0$ below the wall, outside the fluid domain. Superposition with the stagnation flow gives
$$
\boxed{w(z)=\frac12z^2-\frac{i\Gamma}{2\pi}\log(z-z_0)
+\frac{i\Gamma}{2\pi}\log(z-\overline z_0).}
$$
For real $z=x$, the distances $|x-z_0|$ and $|x-\overline z_0|$ agree. The two vortex contributions to the <stream function> are $-\Gamma\log|x-z_0|/(2\pi)$ and $+\Gamma\log|x-\overline z_0|/(2\pi)$, which cancel. The background <stream function> is also zero there, so $y=0$ remains an impermeable streamline. The image changes the boundary condition without introducing another singularity in the upper fluid region.
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