Blasius force from a Laurent coefficient at infinity ($F_x-i{F}_y=-\pi \rho [z^{-1}]q(z{)}^2$)

For a two-dimensional steady ideal flow outside a body, let $q=w^{\prime }$ be the complex velocity. Suppose $q$ is holomorphic in the exterior and $q^2$ has a Laurent series near infinity, with coefficient $a_{-1}$ of $z^{-1}$. Orient the body contour counterclockwise and assume the usual impermeability and Bernoulli hypotheses of the Blasius force theorem. Deforming the contour to a large circle gives $\oint q^{2}dz=2\pi i{a}_{-1}$. Substitution in Blasius' formula proves $F_x-i{F}_y=-\pi \rho a_{-1}$.

In particular, $q(z)=q_{\infty }+O(z^{-2})$ makes the force zero: its square has no reciprocal term. This criterion handles branch-cut obstacles by applying the contour in their analytic exterior, rather than separately assigning forces to endpoint singularities.