= Solution
Assume $q(z)=w'(z)$ is single-valued and a <holomorphic function> between $C$ and a sufficiently large enclosing <circle> $|z|=R$. There are no fluid singularities in that region, so <Cauchy's integral theorem> permits <contour deformation>:
$$
\Gamma+iM=\oint_Cq(z)\,dz=\oint_{|z|=R}q(z)\,dz.
$$
Use the exterior <Laurent series> about zero, valid for $|z|$ larger than every finite singularity:
$$
q(z)=\sum_{n=-\infty}^{\infty}a_nz^n,\qquad a_{-1}=b_1.
$$
It converges uniformly on the chosen <circle>, permitting termwise <contour integration>. For integer $n$, parameterization by $z=Re^{i\theta}$ gives
$$
\oint_{|z|=R}z^n\,dz=iR^{n+1}\int_0^{2\pi}e^{i(n+1)\theta}\,d\theta
=\begin{cases}2\pi i,&n=-1,\\0,&n\ne-1.\end{cases}
$$
Only the $z^{-1}$ coefficient survives, proving
$$
\boxed{\Gamma+iM=2\pi i b_1.}
$$
Thus if $b_1=\alpha+i\beta$, then $\Gamma=-2\pi\beta$ and $M=2\pi\alpha$. A <uniform flow> contributes a constant to $q$ and changes neither integral. The <Laurent series> here is an exterior expansion; it need not converge in a punctured neighborhood of zero if other singularities intervene.
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