A gas composed of identical isotropic molecules has a potential energy of interaction between pairs of particles that depends only on their separation distance. The pair potential is encoded in the virial coefficients of the virial equation of state for a gas. The complete iterative inversion method is a technique employed in an attempt to recover the pair potential from the second virial coefficient. Implicit in the complete iterative inversion method is the requirement that various mathematical expressions are meaningful: improper integrals converge, derivatives exist, etc.We provide a mathematical framework in which all these implicit assumptions are valid. We show that the complete iterative inversion method cannot recover the pair potential even if the target potential and the initial estimate are infinitely differentiable.