Relation between effective couplings for asymptotically free models

For asymptotically free models with two independent couplings asymptotic expansions are constructed which express one effective coupling in terms of the other. The expansions involve powers (including fractional or irrational exponents) and logarithms. All orders of the β-functions are taken into account. The expansions found are complete in the sense that they represent solutions (exact to any order) which generalize all the solutions obtained with the β-functions approximated to second order. It is shown that higher orders are relevant since it is not possible in general to reparametrize the system such that the β-functions become polynomials of the coupling parameters. The simplifications in case of supersymmetric models are discussed. (orig.)