a function is analytic in a sector of opening $\alpha>\tfrac{\pi}{k}$, which is the situation we have above, the map between the analytic function and the asymptotic series of order $k$ is injective (but not surjective) [@balser]. This means that all information about the exact function is contained in the asymptotic series.

An asymptotic series of $\tilde{F}(s)$ in the scale parameter can be obtained doing the perturbative expansion in the coupling and then inserting the running coupling constant. Assume that $\tilde{F}$ is given as a formal perturbative series, which we write as $\tilde{F}(s,g)= \sum_n \hat{F}_n(s) g^n$. The Callan-Symanzik equation states that a theory is invariant, i.e. the correlators are invariant, under a scale transformation if the couplings change according to the renormalisation group. A scaled quantum field theory can therefore equivalently be described by a theory on the same scale, but with couplings changed according to the renormalisation group. In this way scale dependence of a theory can be moved into the running coupling $\bar{g}(s)$. Moving all scale dependence into the running coupling we get an asymptotic series in the scale $\tilde{F}(s=1,\bar{g}(s))=\sum \tilde{F}_n s^n\ $ [^2].

The examples we consider below have $\alpha=\pi-\epsilon'$, where $\epsilon'\ll 1$, hence from the constraint $\alpha>\tfrac{\pi}{k}$: 