an expansion in small $ \tau  $ (and therefore in small $ q $) corresponds to an expansion in small $ y $ of all the quantities like $ f(x,y) $ and $ m(x,y) $; the appearance of integrals of the form $ \int Pfdy $where $ P(x,y)\sim \exp (y^{2}/x) $ generates asymptotic expansions since the Taylor expansions of $ f(x,y) $ and $ m(x,y) $ in powers of $ y $ do not converge on the whole real axes. These arguments can be very useful in practice to guess the position of the singularities of the Borel transform if one want to sum the expansions through a conformal mapping [@Pbook]. For instance in the expression of the free energy appear integrals of the following form:$$\label{as2}
\int ^{+\infty }_{-\infty }\frac{dz}{\sqrt{2\pi \tau }}e^{-\frac{z^{2}}{2\tau }}\ln \cosh (z)$$ The singularities of the Borel transform of the previous integral are located on a cut running from $ -\infty  $ to $ -\pi ^{2}/8 $ and a possible guess is that this be also the singularity structure of the Borel transform of the free energy. This guess is supported by the analysis of the series expansions.

The analytical results have been compared with numerical solutions 