Approximations of Functions With Essential Singularities with Applications to Painlevé's First Transcendent

Abstract

In this work we develop an algorithmic procedure for associating a function defined on the Riemann surface of the $\log$ to given asymptotic data from a function at an essential singularity. We do this by means of rational approximations (Padé approximants) used in tandem with Borel-Écalle summation. Our method is capable of handling situations where classical methods either do not work or converge very slowly eg. We provide a general outline of the procedure and then apply it to generating approximate tritronquée solutions to Painlevé's first equation (${\text{P}}_{\text{I}}$). Our approximations (including ${\text{P}}_{\text{I}}$) are written as a finite linear combination of exponential integrals ${\text{Ei}}^{+}$. Furthermore, we have explicit rational approximations for each ${\text{Ei}}^{+}$ and thus for the approximation as a whole. In addition to rational approximations of ${\text{P}}_{\text{I}}$, we provide the first hundred or so poles of a tritronquée solution with essentially arbitrary accuracy which is dependent upon the order of Padé used.