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

We use a recently developed method [1] and [2] to determine approximate expression for tritronquée solution for P-1: y″+6y2−x=0y″+6y2−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=−770766323285=−2.3841687675⋯ to within 5×10−65×10−6 accuracy, in agreement with previous numerical calculations [6].