A tritronquée solution is one of five distinguished solutions of Painlevé I, the equation defining the first of the six Painlevé transcendents,
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Each solution is a meromorphic function of the complex variable 
. Near infinity, the rays 
for 
, 
, 
divide the complex plane
into five sectors. Crossing one of these boundaries can change exponentially small
terms through the Stokes phenomenon. Generic
solutions have poles accumulating toward all five boundaries,
while a tritronquée is asymptotically free of poles
in four consecutive sectors and therefore admits an asymptotic
series there (Boutroux 1913, 1914). The five choices are related by the fivefold
rotational symmetry 
for 
, 1, ..., 4 (Costin et al. 2014).
One real member is selected by 
as 
. Under the rescaling 
and 
, the equation becomes
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The selected solution then satisfies 
as 
. Its first real pole occurs
at 
,
corresponding to 
.
More precisely, the approximation 
satisfies 
. Adali and Tanveer (2016)
also constructed an explicit piecewise analytic
approximation on a domain avoiding the pole,
with rigorous uniform bounds 
and 
for the error and its derivative.
Ribes Metidieri et al. (2026) used this approximation in the analysis of a
slow passage through a fold catastrophe.
." Duke Math. J. 163, 665-704, 2014.