Rigorous analytical approximation of tritronquée solution to Painlevé-I and the first singularity

Abstract We use a recently developed method [1] , [2] to determine approximate expression for tritronquee solution for P-1: y ″ + 6 y 2 − x = 0 in a domain D with rigorous bounds. In particular we rigorously confirm the location of the closest singularity from the origin to be at x = − 770766 323285 = − 2.3841687675 ⋯ to within 5 × 10 − 6 accuracy, in agreement with previous numerical calculations [6] .