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The Coulomb-nuclear phase shift $\delta^{(cs)}_l$, $\cot\delta^{(cs)}_l$ anda finite limit of the nuclear part $\Delta_l(k)$ of the effective-rangefunction (ERF) are derived for an arbitrary orbital momentum $l$ when energy$E\rightarrow0$. It is proved that $\cot\delta^{(cs)}_l$ has an essentialsingularity at zero energy, but $\Delta_l(k)$ does not. The explicit finitelimit of $\Delta_l(0)$ is found. The property of $\Delta_l(k)$ as a meromorphicfunction makes possible the analytical continuation of a re-normalizedscattering amplitude from the physical energy region to a bound state pole.Then the asymptotic normalization coefficients (ANC) can be deduced fromexperimental phase-shift data and applied to radiative capture processes whichare important in nuclear astrophysics for new elements creation. Our resultsare in agreement with the results published for $S$ wave scattering.

An essential singularity of the cotangent of the Coulomb-nuclear phase  shift, and a finite limit of the nuclear part of the effective-range function  derived at zero energy