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We develop the theory of a new type of asymptotic expansions for functions oftwo variables the coefficients of which contain functions of one of thevariables as well as functions of the quotient of these two variables. Thesecombined asymptotic expansions (DAC) are particularly well suited for thedescription of solutions of singularly perturbed ordinary differentialequations in the neighborhood of turning points. We describe the relations withthe method of matched asymptotic expansions and with the classical compositeasymptotic expansions used for boundary layers. We present a result ofRamis-Sibuya type that proves the existence of these DAC and provides Gevreyestimates. Three applications are given, two of which depend on Gevreyestimates for the DAC. In our article, we only apply the theory to scalarordinary differential equations, but we are convinced that they will be veryuseful for systems of differential equations and other types of functionalequations as well.

D\'eveloppements asymptotiques combin\'es et points tournants d'\'equations diff\'erentielles singuli\`erement perturb\'ees