Effective range expansion for the pion-pion system

For small value of the scattering length a(vertical strokeavertical stroke approx.= 0) the effective range function kappa cot delta has a pole for small k2. In such a case it is advantageous to study an expression of the function k-1 tan delta which is expected to yield a linear dependence on k2 for small k2. An expansion for the function k-1 tan delta is derived and is applied to the pion-pion system to extract the S-wave scattering lengths and effective ranges satisfactorily from phase shifts, where the usual effective range expansion fails because of the occurrence of subthreshold poles in kappa cot delta. (orig.)