The uniform asymptotic method "saddle point near an end point"revisited

We continue the program initiated in [Lopez & all, 2009-2011] to simplify asymptotic methods for integrals: in this paper we revise the uniform method "saddle point near an end point". We obtain a more systematic version of this uniform asymptotic method where the computation of the coefficients of the asymptotic expansion is remarkably simpler than in the classical method. On the other hand, as in the standard method, the asymptotic sequence is given in terms of parabolic cylinder functions. New asymptotic expansions of the confluent hypergeometric functions M(c, x/a + c +1, x) and U(c, ax + c + 1, x) for large x, c fixed, uniformly valid for a is an element of (0, infinity), are given as an illustration.