$k=1+\delta$ for some $\delta>0$. We will generally try to minimise the order $k$ ($k\rightarrow
1$), thus maximising the analytic sector and the convergence of the Borel transform.

Defining $\tilde{F}(s)$ for the central charge
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As the first example we want to calculate the infra-red central charge of a 2 dimensional unitary quantum field theory. The renormalisation group flow of the central charge is governed by Zamolodchikov’s $c$-theorem [@zamolodchikov]. This states that for a unitary and renormalisable quantum field theory in 2 dimensions there exists a function which is monotonically decreasing along the renormalisation group flow, and which is stationary only for conformally invariant theories where it takes the value of the Virasoro central charge. The $c$-theorem implies that the infra-red limit, where the scale goes to infinity, and the ultra-violet limit, where the scale vanishes, are fixed points of the renormalisation group. In 2 dimensions scale invariance implies conformal invariance so in these scaling limits we have a conformal field theory characterised by the central charge. We define the energy-momentum tensor as in [@cappelli] $$\label{Tdef}
  \langle T_{\mu \nu}(x)\rangle = \frac{2V}{\sqrt{g(x)}}
  \frac{\delta W[g]}{\delta g^{\mu\nu}},$$ where $V=\mbox{Vol}(S^{n-1})=2 \pi$ in 2D and $W[g]$ is the effective action. The correlator $\langle T_{zz}(z)T_{zz}(0)\rangle$ gives the 