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In this paper, we present explicit and computable error bounds for theasymptotic expansions of the Hermite polynomials with Plancherel--Rotach scale.Three cases, depending on whether the scaled variable lies in the outer oroscillatory interval, or it is the turning point, are considered separately. Weintroduce the "branch cut" technique to express the error terms as integrals onthe contour taken as the one-sided limit of curves approaching the branch cut.This new technique enables us to derive simple error bounds in terms ofelementary functions. We also provide recursive procedures for the computationof the coefficients appearing in the asymptotic expansions.

Error bounds for the asymptotic expansions of the Hermite polynomials