A better approximation, which accounts for the variation in frequency, is given by

The Plancherel-Rotach asymptotics method, applied to Hermite polynomials, takes into account the uneven spacing of the zeros near the edges. It makes use of the substitution with which one has the uniform approximation

Similar approximations hold for the monotonic and transition regions. Specifically, if then

while for with complex and bounded, the approximation is

where

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