Kinetically dominated curved universes: Logolinear series expansions

We develop a method for computing series expansions out of a singularity for solutions to ordinary differential equations when the asymptotic form contains both linear and logarithmic terms. Such situations are common in primordial cosmology when considering expanding out of a singularity in a pre-inflationary phase of the universe. We develop mathematical techniques for generating these expansions, and apply them to polynomial and Starobinsky inflationary potentials. This paper is the first in a programme of work on kinetically dominated curved universes, for which such power series are essential. Code for analytic and numerical computation of logolinear series is provided on GitHub.