  We develop the theory of a new type of asymptotic expansions for functions of
two variables the coefficients of which contain functions of one of the
variables as well as functions of the quotient of these two variables. These
combined asymptotic expansions (DAC) are particularly well suited for the
description of solutions of singularly perturbed ordinary differential
equations in the neighborhood of turning points. We describe the relations with
the method of matched asymptotic expansions and with the classical composite
asymptotic expansions used for boundary layers. We present a result of
Ramis-Sibuya type that proves the existence of these DAC and provides Gevrey
estimates. Three applications are given, two of which depend on Gevrey
estimates for the DAC. In our article, we only apply the theory to scalar
ordinary differential equations, but we are convinced that they will be very
useful for systems of differential equations and other types of functional
equations as well.
