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We consider the second order linear differential equation where Λ is a large complex parameter and g is a continuous function. In previous works we have considered the case α ∈ (−∞, 2] and designed a convergent and asymptotic method for the solution of the corresponding initial value problem with data at t = 0. In this paper we complete the research initiated in those works and analyze the remaining case α ∈ (2, ∞). We use here the same fixed point technique; the main difference is that for α ∈ (2, ∞) 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 Λ (not of Poincare-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.

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