is moved into the coupling using the running coupling so that $F_{\nu}(s)$ and $F_{\eta}(s)$ are given as asymptotic series in the scale parameter $s$, and we show their analyticity. In this example the actual calculation of the physical quantities $\nu$ and $\eta$ is different form the minimal models case, as we cannot use the same approximation. The reason is that in this case we use higher order perturbative expressions (5,6 and 7 loops) and one cannot find an exact expression for the perturbative running coupling (unlike the one loop case). The infra-red values for $\nu, \eta$ are therefore obtained from the limiting value of the Borel transform, and to calculate this we use a conformal mapping and Padé approximation which is done in the appendix. The estimates we obtain for $\nu, \eta$ are within the errors of other more elaborate approximations.

The main results are that we can use analyticity in the scale parameter to express the infra-red physics as an integral in the ultra-violet region, the infra-red limit is then obtained taking the limit in the Borel transformed quantity. We show how this works in two examples with well known infra-red limits.

The outline is as follows: In the 