The purpose of these lecture notes is to develop a theory of asymptotic expansions for
functions involving two variables while at the same time using functions involving one
variable and functions of the quotient of these two variables. Such composite asymptotic
expansions (CAsEs) are particularly well-suited to describing solutions of singularly perturbed
ordinary differential equations near turning points. CAsEs imply inner and outer expansions
near turning points. Thus our approach is closely related to the method of matched asymptotic
expansions. CAsEs offer two unique advantages however. First they provide uniform expansions
near a turning point and away from it. Second a Gevrey version of CAsEs is available and
detailed in the lecture notes. Three problems are presented in which CAsEs are useful. The
first application concerns canard solutions near a multiple turning point. The second
application concerns so-called non-smooth or angular canard solutions. Finally an
Ackerberg-O'Malley resonance problem is solved.
