[*for all $\epsilon>0$ the Helmholtz-Laplace equation admits a solution satisfying both the boundary conditions on the circle and the velocity condition at infinity*]{}, obtained by considering elementary solutions that are linear combinations of solutions of the Laplace equation and solutions of the Helmholtz equation.

Note that this approach and our [*ad hoc*]{} scheme are linked by the relationship $$\epsilon^2 = \lambda R_e^{2\beta}$$ The determination of $\lambda$ and $\beta$ are interesting challenges. We shall see in Section 8 that there are good theoretical grounds for setting $\beta=1$. For now we will leave both parameters general.

The case of asymptotically uniform flow past a circle
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We now consider the well-trodden route to the analysis of a stream function associated with a flow that is uniform at infinity and satisfies appropriate boundary conditions on $r=1$. Using polar coordinates, we therefore want $$\psi(r,\theta) \sim r \sin\theta  \ \ {\rm as}\ \  r \rightarrow \infty$$ and $$\psi = 0 = \frac{\partial \psi}{\partial r}\ \  {\rm on}\ \  r=1 \label{circlebc}$$ We build the solution for $\psi_0$ and $\psi_1$ as follows, under the assumption that $\psi_0$ satisfies the Laplace equation and $\psi_1$ the Helmholtz conditions. Any sum of the two will satisfy the Laplace-Helmholtz equation. Given 