= Solution
The outgoing wave is singular at the origin, so its value there cannot be inserted without <regularization>. At a short radial distance $\delta>0$, with $k\delta\ll1$,
$$
\psi_k(\delta)=1+\frac{f_ke^{ik\delta}}{\delta}+O(k\delta)
=1+f_k\left(\frac1\delta+ik\right)
+O(k\delta)+O(f_kk^2\delta).
$$
The incident factor tends to one independently of the chosen approach direction; the singular part is isotropic. The discrete-state equation becomes, to the retained short-distance order,
$$
\boxed{(E_k-\epsilon_0)d_k
=g\left[1+f_k\left(\frac1\delta+ik\right)\right].}
$$
Keeping only $1/\delta$ and discarding $ik$ would lose the outgoing-wave imaginary part and produce an amplitude inconsistent with scattering <unitarity>.
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