Approximations of Functions With Essential Singularities with Applications to Painlev\'e's First Transcendent

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 (Pad\'e approximants) used in tandem with Borel-\'Ecalle summation. Our method is capable of handling situations where classical methods either do not work or converge very slowly eg. {DunLutz}. We provide a general outline of the procedure and then apply it to generating approximate tritronqu\'ee solutions to Painlev\'e's first equation (PI). Our approximations (including PI) are written as a finite linear combination of exponential integrals Ei⁺. Furthermore, from arXiv:2210.17502 we have explicit rational approximations for each Ei⁺ and thus for the approximation as a whole. In addition to rational approximations of PI, we provide the first hundred or so poles of a tritronqu\'ee solution with essentially arbitrary accuracy which is dependent upon the order of Pad\'e used.