= Finite-energy condition for planar exterior potential flow
{title2=$q(z)=O(z^{-2})$}
For a planar <potential flow> outside a bounded body, suppose its single-valued <complex velocity> $q=u-iv$ is holomorphic near infinity and tends to zero. Its exterior <Laurent series> begins $q=b_1/z+O(z^{-2})$. The outward <volume flux> $Q$ and counterclockwise <circulation> $\Gamma$ satisfy $\Gamma+iQ=2\pi i b_1$. Thus both vanish precisely when the leading term is absent. A nonzero $b_1$ gives $|q|^2\sim|b_1|^2/r^2$ and logarithmically divergent exterior <kinetic energy>. When $b_1=0$, $q=O(r^{-2})$ and the far-field energy integral converges. Zero <circulation> alone does not exclude the radial source field $\nabla\log r$.
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