Second order differential equations with an irregular singularity at the origin and a large parameter: convergent and asymptotic expansions

We consider the second order linear differential equation

where is a large complex parameter and is a continuous function. In previous works we have considered the case and designed a convergent and asymptotic method for the solution of the corresponding initial value problem with data at . In this paper we complete the research initiated in those works and analyze the remaining case . We use here the same fixed point technique; the main difference is that for the convergence of the method requires that the initial datum is given at a point different from the origin; for convenience we choose the point at the infinity. We obtain a sequence of functions that converges to the unique solution of the problem. This sequence has also the property of being an asymptotic expansion for large (not of Poincar\’e-type) of the solution of the problem. The generalization to non-linear problems is straightforward. An application to a quantum mechanical problem is given as an illustration.