 Complete-flyby gravitational impulse

Integrate the transverse force along a complete straight path: ${\int }_{-\infty }^{\infty }(1+s^2{)}^{-3/2}ds=2$. This gives the transverse kick. Conservation of relative speed gives longitudinal speed reduction $\Delta v_{\perp }^2/(2V)$ at second order. As an independent check, exact weak hyperbolic scattering has angle $2\arctan \eta$, where $\eta =GM/(b{V}^2)$, and reduction $V(1-\cos (2\arctan \eta ))=2V{\eta }^2/(1+{\eta }^2)$. The small-$\eta$ coefficient agrees. Keeping only one half of the encounter in the transverse kick makes this quadratic reduction four times too small.