We have $$ \psi_k(r) = e^{ikr} + f(\theta) \frac{e^{ikr}}{r} $$ with $$ f(\theta) = \sum_l (2l+1)f_l(\theta) P_l(\cos(\theta)) $$ where the partial wave scattering amplitude is given by $$ f_l(\theta) = \frac{1}{k}\frac{(S_l(k)-1)}{2i} = \frac{1}{k}\sin\delta_l(k) e^{i\delta_l(k)} $$ With Eulers formula for the cotangent, this can also be written as $$ f_l(\theta) = \frac{1}{k}\frac{1}{\cot \delta_l(k) - i}. $$