$\tilde{F}(s)$ in (\[F2\]) it follows that $\tilde{F}(s)$ has a limit value $\tilde{F}(s)\rightarrow d$ for $s\rightarrow 0$ from $S$. From (\[Fgraense\]) it follows that $d=c_{UV}$, and we define $\tilde{F}(0)=c_{UV}$. It also follows from (\[F2\]) that $\tilde{F}(s)$ has a limit value for $|s|\rightarrow \infty$ from $S$: $\tilde{F}(s)\rightarrow d'$ and (\[Fgraense\]) again sets $d'=c_{IR}$.

The approximation for $c_{IR}$
------------------------------

We have the following representation of $c_{IR}$ and $c_{UV}$ $$\begin{aligned}
c_{IR} &=\frac{1}{2\pi i}\left( \int_{C_0}ds \frac{e^{\rho /s}}{s}F(s)+
\int_{C_1}ds \frac{e^{\rho /s}}{s}F(s)  \right)=I_{IR}(\rho)+cut(\rho)
\label{align2} \\
c_{UV} &=\frac{-1}{2\pi i}\left(\int_{C_2}ds \frac{e^{\rho s}}{s}F(s)+
\int_{C_1}ds \frac{e^{\rho s}}{s}F(s)  \right)=I_{UV}(\rho)+cut(\rho), \label{align1} \end{aligned}$$ where in both cases the contribution from the cut is rapidly decreasing in $\rho$. Note that in these relations the integral is performed in the opposite scaling limit of the quantity we calculate. We will here concentrate on as perturbation theory can be applied to $F(s)$ when $s\in C_0$, we will call $I_{IR}(\rho)=I(\rho)$.

We can choose coordinates in the coupling constant space so that the ultra-violet fixed point corresponds to $\bar{g}(s)=0$. In the ultra-violet limit where $s\rightarrow 0$ we may describe $F(s)$ by perturbation theory as $\bar{g}(s)\rightarrow 0$. The nth order perturbative approximation of $F(s)$ is denoted by $F_n(s)$ and the corresponding integral by $I_n(\rho)$. In the limit of large $\rho$ 