## Composite Asymptotic ExpansionsThe 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. |

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### Contents

1 | |

General Study | 17 |

Gevrey Theory | 42 |

Chapter
4 A Theorem of RamisSibuya Type | 63 |

Chapter
5 Composite Expansions and Singularly Perturbed Differential Equations | 81 |

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analogous analytically continued antiderivative argx assumption canard solutions canard value Cauchy inequalities change of variable class C1 classical coefficients Composite Asymptotic Expansions composite expansion composite formal series composite series consistent good covering constant Corollary 5.16 defined for n e denote disk domain estimate existence expansion of Gevrey exponentially close exponentially small formal solution formula Fruchard function defined functions g Gevrey order holomorphic function implicit function theorem implies initial condition inner equation inner expansion integration Lemma Math n e S2 neighborhood notation obtain Œa;b ordinary differential equations outer expansions parameter polynomial proof of Lemma proof of Theorem Proposition 2.16 prove Remark resonant solution resp result Riccati equation satisfies Sch¨afke Sect sector Similarly singularly perturbed slow curve Taylor expansion tends term Theorem 6.5 turning point unique solution wronskian x e V(n x-sector yo(x