  In this paper, we present different proofs of very recent results on the
necessary as well as sufficient conditions on the decrease of the potential at
infinity for the validity of effective range formulas in 3-D in low energy
potential scattering (Andr\'e Martin, private communication, to appear. See
Theorem 1 below). Our proofs are based on compact formulas for the
phase-shifts. The sufficiency conditions are well-known since long. But the
necessity of the same conditions for potentials keeping a constant sign at
large distances are new. All these conditions are established here for
dimension 3 and for all angular momenta $\ell \geq 0$.
