Second virial coefficient in one dimension as a function of asymptotic quantities

A result from Dodd and Gibbs (J. Math. Phys., 15, 41 (1974)) for the second virial coefficient of particles in one dimension, subject to delta-function interactions, has been obtained by direct integration of the wave functions. It is shown that this result can be obtained from a phase shift formalism, if one also includes the contribution of oscillating terms. The result is important in work to follow, for the third virial coefficient, for which a similar formalism is being developed. We examine a number of fine points in the quantum mechanical formalisms.