English: An illustration of the three approximations to the probabalists Hermite polynomials, each accurate in a different regime and plotted so that the $y$-axis is a modified log scale. The three regmies are: $x=2\sqrt{n}\cos(\alpha) < 2\sqrt{n}$ for small $x$ values, $x=2\sqrt{n}\cosh(\gamma) > 2\sqrt{n}$ for large $x$ values, and $x=2\sqrt{n} + n^{-1/6}u$ for $x$ values on the edge near $x=2\sqrt{n}$. Notice from the zoomed insert that the small-x and large-x approximations are both singular at the edge $x=2\sqrt{n}$, and this is the regime where the edge approximation with the Airy function is accurate.