We construct the analytic solutions of a family of linear singularly perturbed Cauchy problems under the action of linear fractional transforms and relate them with an asymptotic formal solution via Gevrey asymptotic expansions. The construction of the analytic solutions is based on different Laplace-like integrals along paths of different nature which arise from the appearance of domains in the complex plane which remain intimately related to Lambert W function. A deep knowledge on the behavior of W Lambert function is needed to perform adecquate deformation paths of the analytic solutions.