Addressing the question how to sum a power series in one variable when it diverges that is
how to attach to it analytic functions the volume gives answers by presenting and comparing
the various theories of k-summability and multisummability. These theories apply in particular
to all solutions of ordinary differential equations. The volume includes applications examples
and revisits from a cohomological point of view the group of tangent-to-identity germs of
diffeomorphisms of C studied in volume 1. With a view to applying the theories to solutions of
differential equations a detailed survey of linear ordinary differential equations is provided
which includes Gevrey asymptotic expansions Newton polygons index theorems and Sibuya's proof
of the meromorphic classification theorem that characterizes the Stokes phenomenon for linear
differential equations. This volume is the second in a series of three entitled Divergent
Series Summability and Resurgence. It is aimed at graduate students and researchers in
mathematics and theoretical physics who are interested in divergent series Although closely
related to the other two volumes it can be read independently.
