Outgoing near-radial Schwarzschild light deflection ($\Delta \varphi =\psi /\sqrt{1-2GM/(R{c}^2)}+O({\psi }^3)$)

A Schwarzschild static observer at areal radius $R$ emits an outward null geodesic at a small angle $\psi$ to the radial direction. Its impact parameter is $b=R\sin \psi /\sqrt{f}$, and the accumulated azimuth to infinity is ${\int }_R^{\infty }bdr/[r^2\sqrt{1-f(r)b^2/r^2}]=b/R+O(b^3/R^3)$. Thus weak curvature magnifies a narrow outward beam angle by $1/\sqrt{f(R)}$, rather than producing the full infinity-to-infinity deflection $4GM/(b{c}^2)$.