the infra-red limit $F_{IR}=\lim_{|s|\rightarrow
  \infty}F(s)$ as an integral $$\label{secondeq}
  F_{IR}= \frac{1}{2\pi i}\left( \int_{C_0}+\int_{C_1}\right)\frac{ds}{s}e^{\rho/s}F(s)
\nonumber$$ over the infinitesimal circle $C_0$ and the cut $C_1$.

We show below that the integral over the cut $C_1$ vanishes for large $\rho$, so that the infra-red limit $F_{IR}$ then is given as an integral in the UV region over $C_0$ where perturbation theory applies. We call this integral $I(\rho)$ and get that $F_{IR}=\lim_{\rho \rightarrow \infty}I(\rho)$.

$I(\rho)$ is shown to be the Borel transform of $F(s)$ of order $k$, where $k$ is related to the way we analytically continue in the complex scale parameter. The Borel transform has also been applied to perturbation theory in [@LGZJ] where the Borel transform in the coupling is used. In this case assumptions (justified by details of the dynamics) have to be made about the poles of the Borel transform. The essential difference compared with our approach is that we transform with respect to the scale parameter. The analytic properties of Green functions, which are needed to define the Borel transform, in this case follow directly from general principles of quantum field theory.

We will use this method on two examples where the infra-red limit of the physical quantity $F(s)$ 