= Solution
The <Milne-Thomson circle theorem> says that if a complex potential $f(z)$ has no singularities on or inside $|z|=a$, then a circular impermeable boundary can be introduced by taking
$$
W(z)=f(z)+f^*(a^2/z),\qquad f^*(w)=\overline{f(\overline w)}.
$$
The new singularities are the reflected ones inside the circle. On $|z|=a$, $a^2/z=\overline z$, so $W(z)=f(z)+\overline{f(z)}$ is real; the <stream function> is constant and there is no normal velocity. At infinity the added term tends to $f^*(0)$, a harmless potential constant, so it preserves the far-<field> flow. This proves the theorem with zero added net <circulation> about the cylinder. A logarithmic term may then specify a different cylinder <circulation>.
For $f(z)=-i\Gamma\log(z-b)/(2\pi)$, put $c=a^2/b$. The reflected term is
$$
f^*(a^2/z)=\frac{i\Gamma}{2\pi}\log(a^2/z-b)
=\frac{i\Gamma}{2\pi}\log(z-c)-\frac{i\Gamma}{2\pi}\log z+\text{constant},
$$
with locally consistent logarithm branches. The circle-theorem solution has zero cylinder <circulation>: its image <vortex> at $c$ and its central <vortex> cancel. To impose <circulation> $-\Gamma$, add $i\Gamma\log z/(2\pi)$. The result is
$$
\boxed{w(z)=-\frac{i\Gamma}{2\pi}\log(z-b)
+\frac{i\Gamma}{2\pi}\log(z-c)+\text{constant}.}
$$
It consists of the physical <vortex> and an opposite image <vortex> inside the obstacle. On the circle, $|z-b|=(b/a)|z-c|$, so its <stream function> is indeed constant.
The <complex velocity> is
$$
q(z)=w'(z)=-\frac{i\Gamma}{2\pi}
\left(\frac1{z-b}-\frac1{z-c}\right).
$$
Inside the positively oriented obstacle contour, $q^2$ has only the double pole at $c$. Its <residue> is
$$
\operatorname{Res}_{z=c}q(z)^2
=\left.\frac d{dz}\left[-\frac{\Gamma^2(b-c)^2}{4\pi^2(z-b)^2}\right]\right|_{z=c}
=-\frac{\Gamma^2}{2\pi^2(b-c)}.
$$
The <residue theorem> and the supplied <Blasius force theorem> therefore give
$$
F_x-iF_y=\frac{i\rho}{2}(2\pi i)\left[-\frac{\Gamma^2}{2\pi^2(b-c)}\right]
=\frac{\rho\Gamma^2}{2\pi(b-c)}.
$$
Consequently
$$
\boxed{F_x=\frac{\rho\Gamma^2b}{2\pi(b^2-a^2)},\qquad F_y=0.}
$$
The force points toward the held <vortex>, independent of the sign of its <circulation>.
The residue calculation is the <force on a circular cylinder from a held exterior vortex>.
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