= Solution
The <method of images> places an equal <line source> at $-ia$, outside the physical <upper half-plane>. A convenient <complex potential of a two-dimensional potential flow> is
$$
\boxed{w(z)=\frac Q{2\pi}\left[\log(z-ia)+\log(z+ia)\right],\qquad
w'(z)=\frac{Qz}{\pi(z^2+a^2)}.}
$$
The two logarithms can be combined locally as $(Q/(2\pi))\log(z^2+a^2)$; their branches of the <complex logarithm> affect only the potential convention, not the single-valued <complex velocity>. The <point source> in this two-dimensional problem has the <line source> normalization: its outward <volume flux> is $Q$ per unit span.
Writing $R_-^2=x^2+(y-a)^2$, $R_+^2=x^2+(y+a)^2$, and using $w'=u-iv$, the <velocity field> is
$$
\boxed{u=\frac Q{2\pi}\left(\frac{x}{R_-^2}+\frac{x}{R_+^2}\right),\qquad
v=\frac Q{2\pi}\left(\frac{y-a}{R_-^2}+\frac{y+a}{R_+^2}\right).}
$$
On the wall $y=0$, the vertical components cancel:
$$
\boxed{v(x,0)=0,\qquad u(x,0)=\frac{Qx}{\pi(x^2+a^2)}.}
$$
This is the required <no-penetration boundary condition>. Tangential slip is permitted. At large distance the <velocity> decays as $1/r$, and near $z=ia$ its singular part is exactly that of the specified source; the reflected source is regular throughout the fluid. Away from the injection point, this is an <incompressible flow> and an <irrotational flow>.
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