Asymptotic Expansions and Summability

Pascal Remy is a research associate at the Laboratoire de Mathématiques de Versailles, at the University of Versailles Saint-Quentin (France). His main interest is the theory of summation of divergent formal power series (including Gevrey estimates, summability, multi-summability, and Stokes phenomenon). His research extends to applications such as formal solutions of meromorphic linear differential equations, partial differential equations and integro-differential equations, both linear and nonlinear.

“This work is a valuable source for both researchers in the field and those interested in understanding the transition from formal power series solutions to sectorial, analytic ones for some wide families of PDEs and more general equations.” (Javier Sanz, Mathematical Reviews, June, 2025)“The text are illustrated with many applications to concrete partial differential equations such as the heat equation, the Burgers equation, etc. and also to applications in physics. Moreover, the author provides references of the main results and applications for a deepened reading.” (Alberto Lastra, zbMATH 1555.35003, 2025)

• - Part I Asymptotic expansions.- Taylor expansions.- Gevrey formal power series.- Gevrey asymptotics.- Part II Summability.- k-summability: definition and first algebraic properties.- First characterization of the k-summability: the successive derivatives.- Second characterization of the k-summability: the Borel-Laplace method.- Part III Moment summability.- Moment functions and moment operators.- Moment-Borel-Laplace method and summability.- Linear moment partial differential equations.