= Solution
For the same counterclockwise <line vortex> strength $\kappa$, its local <velocity potential> and <streamfunction> are $\phi=\kappa\theta/(2\pi)$ and $\psi=-\kappa\log r/(2\pi)$. Consequently
$$
w(z)=-\frac{i\kappa}{2\pi}\log z,\qquad w'(z)=-\frac{i\kappa}{2\pi z},\qquad b_1=-\frac{i\kappa}{2\pi}.
$$
Substitution gives
$$
\boxed{2\pi i b_1=\kappa=\Gamma+iM,\qquad M=0,}
$$
in agreement with the direct <circulation> and <volume flux> calculation. The sign of the imaginary coefficient is essential for a positive counterclockwise <circulation>.
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