correct limit when $m \rightarrow \infty$, where $m$ characterises the minimal model. We then show how our method improves upon the perturbative renormalisation group result. If we use the largest domain of analyticity for $F(s)$ apparent from the spectral decomposition there is something of a surprise. The approximation becomes dramatically close to the exact value, which is remarkable as it is only based on one loop perturbation theory.

The other example is $\varphi^4$ theory in three dimensions, where we calculate the critical exponents $\nu$ and $\eta$. $\varphi^4$ theory has been shown to be Borel summable of order one in the perturbative expansion in the coupling constant. The perturbative series will often at most be asymptotic, if for example a theory is ill defined for negative couplings the convergence radius will be zero. The question is then if the perturbative series captures all of the physics when it is only asymptotic. This will depend on the analytic properties of the correlators, and if the theory is Borel summable perturbation theory will contain all information about the theory. We use the propagator to define functions $F_{\nu}, F_{\eta}$ which have the critical exponents $\nu$ and $\eta$ as their infra-red limit. All scale dependence 