 Solution

For an interaction of finite range $b$, higher partial waves are suppressed at $kb\ll 1$ by the centrifugal barrier, away from exceptional higher-wave resonances. Outside the potential, the zero-energy radial S-wave scattering equation is $u^{\prime \prime }=0$, where $\psi =u/r$. Its solution is proportional to $r-a$, so normalization of the constant term givesThe scattering length $a$ is the extrapolated zero of $u(r)$. At nonzero small $k$, this form applies in the overlap region $b\ll r\ll k^{-1}$; it is not the far-field limit at fixed $k$. Equivalently, $k\cot {\delta }_0\to -1/a$ and $f_0\to -a$.