= 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 is
$$
\overline{\frac m{\overline{iz}-a}}=\frac{\bar m}{iz-\bar a}=-\frac{i\bar m}{z+i\bar a}.
$$
Thus
$$
\boxed{F(z)=\frac m{z-a}-\frac{i\bar m}{z+i\bar a}.}
$$
The image is the reflected point $-i\bar a$, which lies in the opposite half-plane when the original <planar potential dipole>[dipole] lies strictly inside the fluid half-plane. The wall itself is a limiting boundary, not an admissible position for the singularity.
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