Weber parabolic cylinder functions

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11: 2.8 Differential Equations with a Parameter

… ►For $J_{\nu }$ and $Y_{\nu }$ see §10.2(ii). … ►Define … ►For two coalescing turning points see Olver (1975a, 1976) and Dunster (1996a); in this case the uniform approximants are parabolic cylinder functions. (For envelope functions for parabolic cylinder functions see §14.15(v)). … ►For further examples of uniform asymptotic approximations in terms of parabolic cylinder functions see §§13.20(iii), 13.20(iv), 14.15(v), 15.12(iii), 18.24. …

12: 13.8 Asymptotic Approximations for Large Parameters

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§13.8(ii) Large $b$ and $z$, Fixed $a$ and $b/z$

… ►For the parabolic cylinder function $U$ see §12.2, and for an extension to an asymptotic expansion see Temme (1978). … ►

§13.8(iii) Large $a$

… ► … ►where $C_{\nu }(a,\zeta )=\cos \left(\pi a\right)J_{\nu }\left(\zeta \right)+\sin \left(\pi a\right)Y_{\nu }\left(\zeta \right)$ and …

13: 14.15 Uniform Asymptotic Approximations

… ►For the Bessel functions $J$ and $Y$ see §10.2(ii), and for the $env$ functions associated with $J$ and $Y$ see §2.8(iv). … ►Here we introduce the envelopes of the parabolic cylinder functions $U(-c,x)$, $\overline{U}(-c,x)$, which are defined in §12.2. For $U(-c,x)$ or $\overline{U}(-c,x)$, with $c$ and $x$ nonnegative, …where $x=X_c$ denotes the largest positive root of the equation $U(-c,x)=\overline{U}(-c,x)$. … ►