 Solution

On the line $y=-x$ one has $\overline{iz}=z$. Hence the reflected term in the printed formula is $\overline{w(z)}$, and the total is real on the line. This proves that its stream function is constant there. The nested conjugation is important: it makes the added term analytic as a function of $z$.

For the planar potential dipole its image contribution isThusThe image is the reflected point $-i\bar{a}$, which lies in the opposite half-plane when the original dipole lies strictly inside the fluid half-plane. The wall itself is a limiting boundary, not an admissible position for the singularity.