= Solution
Multiply the <complex velocity> by the complex displacement:
$$
\frac{dw}{dz}\,dz=(u-iv)(dx+i\,dy)=(u\,dx+v\,dy)+i(u\,dy-v\,dx).
$$
The real term is the <circulation> integrand. With a counterclockwise curve, $\mathbf n\,ds=(dy,-dx)$ is outward, so the imaginary term is the outward <volume flux> integrand. Integrating gives
$$
\boxed{\oint_{C(t)}\frac{dw}{dz}\,dz=\Gamma(t)+iM(t).}
$$
This also fixes the sign of $M$: a positive <line source> has $w'=Q/(2\pi z)$ and the integral is $iQ$, so its outward <volume flux> is $Q$.
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