Editing Hermite polynomials (section)

===Asymptotic expansion===
Asymptotically, as {{math|''n'' → ∞}}, the expansion<ref>{{harvnb|Abramowitz|Stegun|1983|page=508–510}}, [http://www.math.sfu.ca/~cbm/aands/page_508.htm 13.6.38 and 13.5.16].</ref>
<math display="block">e^{-\frac{x^2}{2}}\cdot H_n(x) \sim \frac{2^n}{\sqrt \pi}\Gamma\left(\frac{n+1}2\right) \cos \left(x \sqrt{2 n}- \frac{n\pi}{2} \right)</math> 
holds true. For certain cases concerning a wider range of evaluation, it is necessary to include a factor for changing amplitude:
<math display="block">e^{-\frac{x^2}{2}}\cdot H_n(x) \sim \frac{2^n}{\sqrt \pi}\Gamma\left(\frac{n+1}2\right) \cos \left(x \sqrt{2 n}- \frac{n\pi}{2} \right)\left(1-\frac{x^2}{2n+1}\right)^{-\frac14}=\frac{\Gamma(n)}{\Gamma\left(\frac{n}2\right)} \cos \left(x \sqrt{2 n}- \frac{n\pi}{2} \right)\left(1-\frac{x^2}{2n+1}\right)^{-\frac14},</math>
which, using [[Stirling's approximation]], can be further simplified, in the limit, to
<math display="block">e^{-\frac{x^2}{2}}\cdot H_n(x) \sim \left(\frac{2n}{e}\right)^{\frac{n}{2}} \sqrt{2} \cos \left(x \sqrt{2n}- \frac{n\pi}{2} \right)\left(1-\frac{x^2}{2n+1}\right)^{-\frac14}.</math>

This expansion is needed to resolve the [[wavefunction]] of a [[quantum harmonic oscillator]] such that it agrees with the classical approximation in the limit of the [[correspondence principle]].

A better approximation, which accounts for the variation in frequency, is given by
<math display="block">e^{-\frac{x^2}{2}}\cdot H_n(x) \sim \left(\frac{2n}{e}\right)^{\frac{n}{2}} \sqrt{2} \cos \left(x \sqrt{2n+1-\frac{x^2}{3}}- \frac {n\pi}{2} \right)\left(1-\frac{x^2}{2n+1}\right)^{-\frac14}.</math>

A finer approximation,<ref>{{harvnb|Szegő|1955|p=201}}</ref> which takes into account the uneven spacing of the zeros near the edges, makes use of the substitution 
<math display="block">x = \sqrt{2n + 1}\cos(\varphi), \quad 0 < \varepsilon \leq \varphi \leq \pi - \varepsilon,</math> 
with which one has the uniform approximation
<math display="block">e^{-\frac{x^2}{2}}\cdot H_n(x) = 2^{\frac{n}{2}+\frac14}\sqrt{n!}(\pi n)^{-\frac14}(\sin \varphi)^{-\frac12} \cdot \left(\sin\left(\frac{3\pi}{4} + \left(\frac{n}{2} + \frac{1}{4}\right)\left(\sin 2\varphi-2\varphi\right) \right)+O\left(n^{-1}\right) \right).</math>

Similar approximations hold for the monotonic and transition regions. Specifically, if 
<math display="block">x = \sqrt{2n+1} \cosh(\varphi), \quad 0 < \varepsilon \leq \varphi \leq \omega < \infty,</math> 
then
<math display="block">e^{-\frac{x^2}{2}}\cdot H_n(x) = 2^{\frac{n}{2}-\frac34}\sqrt{n!}(\pi n)^{-\frac14}(\sinh \varphi)^{-\frac12} \cdot e^{\left(\frac{n}{2}+\frac{1}{4}\right)\left(2\varphi-\sinh 2\varphi\right)}\left(1+O\left(n^{-1}\right) \right),</math>
while for <math display="block">x = \sqrt{2n + 1} + t</math> with {{mvar|t}} complex and bounded, the approximation is
<math display="block">e^{-\frac{x^2}{2}}\cdot H_n(x) =\pi^{\frac14}2^{\frac{n}{2}+\frac14}\sqrt{n!}\, n^{-\frac{1}{12}}\left( \operatorname{Ai}\left(2^{\frac12}n^{\frac16}t\right)+ O\left(n^{-\frac23}\right) \right),</math>
where {{math|Ai}} is the [[Airy function]] of the first kind.