# envelope functions

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##### 1: 2.8 Differential Equations with a Parameter
of smallest absolute value, and define the envelopes of $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$ by …These envelopes are continuous functions of $x$, and as $u\to\infty$Again, an alternative way of representing the error terms in (2.8.29) and (2.8.30) is by means of envelope functions. …Define … (For envelope functions for parabolic cylinder functions see §14.15(v)). …
##### 2: 13.21 Uniform Asymptotic Approximations for Large $\kappa$
13.21.1 $M_{\kappa,\mu}\left(x\right)=\sqrt{x}\Gamma\left(2\mu+1\right)\kappa^{-\mu}\*% \left(J_{2\mu}\left(2\sqrt{x\kappa}\right)+\mathrm{env}\mskip-2.0mu J_{2\mu}% \left(2\sqrt{x\kappa}\right)O\left(\kappa^{-\frac{1}{2}}\right)\right),$
13.21.2 $W_{\kappa,\mu}\left(x\right)=\sqrt{x}\Gamma\left(\kappa+\tfrac{1}{2}\right)\*% \left(\sin\left(\kappa\pi-\mu\pi\right)J_{2\mu}\left(2\sqrt{x\kappa}\right)-% \cos\left(\kappa\pi-\mu\pi\right)Y_{2\mu}\left(2\sqrt{x\kappa}\right)+\mathrm{% env}\mskip-2.0mu Y_{2\mu}\left(2\sqrt{x\kappa}\right)O\left(\kappa^{-\frac{1}{% 2}}\right)\right),$
13.21.13 $M_{\kappa,\mu}\left(x\right)=\Gamma\left(2\mu+1\right)\*\left(\frac{e^{2}}{% \kappa^{2}-\mu^{2}}\right)^{\frac{1}{2}\mu}\*\left(\frac{\kappa-\mu}{\kappa+% \mu}\right)^{\frac{1}{2}\kappa}\*\Psi(\kappa,\mu,x)\*\left(J_{2\mu}\left(\sqrt% {\zeta\kappa}\right)+\mathrm{env}\mskip-2.0mu J_{2\mu}\left(\sqrt{\zeta\kappa}% \right)O\left(\kappa^{-1}\right)\right),$
13.21.14 $W_{\kappa,\mu}\left(x\right)=\frac{e^{-\mu\pi\mathrm{i}}}{\pi}\Gamma\left(% \kappa+\mu+\tfrac{1}{2}\right)\*\Gamma\left(\kappa-\mu+\tfrac{1}{2}\right)\*c(% \kappa,\mu)\Psi(\kappa,\mu,x)\*\left(\sin\left(\kappa\pi-\mu\pi\right)J_{2\mu}% \left(\sqrt{\zeta\kappa}\right)-\cos\left(\kappa\pi-\mu\pi\right)Y_{2\mu}\left% (\sqrt{\zeta\kappa}\right)+\mathrm{env}\mskip-2.0mu Y_{2\mu}\left(\sqrt{\zeta% \kappa}\right)O\left(\kappa^{-1}\right)\right),$
13.21.22 $M_{\kappa,\mu}\left(x\right)=\frac{1}{2\pi}\*\Gamma\left(2\mu+1\right)\*\Gamma% \left(\kappa-\mu+\tfrac{1}{2}\right)\*\widehat{c}(\kappa,\mu)\*\widehat{\Psi}(% \kappa,\mu,x)\*\left(\sin\left(\kappa\pi-\mu\pi\right)\mathrm{Ai}\left(\kappa^% {\frac{2}{3}}\widehat{\zeta}\right)+\cos\left(\kappa\pi-\mu\pi\right)\mathrm{% Bi}\left(\kappa^{\frac{2}{3}}\widehat{\zeta}\right)+\mathrm{envBi}\left(\kappa% ^{\frac{2}{3}}\widehat{\zeta}\right)O\left(\kappa^{-1}\right)\right),$
##### 4: 13.20 Uniform Asymptotic Approximations for Large $\mu$
13.20.16 $W_{\kappa,\mu}\left(x\right)=\left(\tfrac{1}{2}\mu\right)^{-\frac{1}{4}}\*% \left(\frac{\kappa+\mu}{e}\right)^{\frac{1}{2}(\kappa+\mu)}\*\Phi(\kappa,\mu,x% )\*\left(U\left(\mu-\kappa,\zeta\sqrt{2\mu}\right)+\mathrm{env}\mskip-1.0mu U% \left(\mu-\kappa,\zeta\sqrt{2\mu}\right)O\left(\mu^{-\frac{2}{3}}\right)\right),$
13.20.17 $M_{\kappa,\mu}\left(x\right)=\left(8\mu\right)^{\frac{1}{4}}\*\left(\frac{2\mu% }{e}\right)^{2\mu}\*\left(\frac{e}{\kappa+\mu}\right)^{\frac{1}{2}(\kappa+\mu)% }\*\Phi(\kappa,\mu,x)\*\left(U\left(\mu-\kappa,-\zeta\sqrt{2\mu}\right)+% \mathrm{env}\mskip-1.0mu \overline{U}\left(\mu-\kappa,\zeta\sqrt{2\mu}\right)O% \left(\mu^{-\frac{2}{3}}\right)\right),$
13.20.18 $W_{\kappa,\mu}\left(x\right)=\left(\tfrac{1}{2}\mu\right)^{-\frac{1}{4}}\*% \left(\frac{\kappa+\mu}{e}\right)^{\frac{1}{2}(\kappa+\mu)}\*\Phi(\kappa,\mu,x% )\*\left(U\left(\mu-\kappa,\zeta\sqrt{2\mu}\right)+\mathrm{env}\mskip-1.0mu % \overline{U}\left(\mu-\kappa,-\zeta\sqrt{2\mu}\right)O\left(\mu^{-\frac{2}{3}}% \right)\right),$
13.20.19 $M_{\kappa,\mu}\left(x\right)=\left(8\mu\right)^{\frac{1}{4}}\*\left(\frac{2\mu% }{e}\right)^{2\mu}\*\left(\frac{e}{\kappa+\mu}\right)^{\frac{1}{2}(\kappa+\mu)% }\*\Phi(\kappa,\mu,x)\*\left(U\left(\mu-\kappa,-\zeta\sqrt{2\mu}\right)+% \mathrm{env}\mskip-1.0mu U\left(\mu-\kappa,-\zeta\sqrt{2\mu}\right)O\left(\mu^% {-\frac{2}{3}}\right)\right),$
##### 5: 14.15 Uniform Asymptotic Approximations
14.15.11 $\mathsf{P}^{-\mu}_{\nu}\left(\cos\theta\right)=\frac{1}{\nu^{\mu}}\left(\frac{% \theta}{\sin\theta}\right)^{1/2}\left(J_{\mu}\left(\left(\nu+\tfrac{1}{2}% \right)\theta\right)+O\left(\frac{1}{\nu}\right)\mathrm{env}\mskip-2.0mu J_{% \mu}\left(\left(\nu+\tfrac{1}{2}\right)\theta\right)\right),$
14.15.12 $\mathsf{Q}^{-\mu}_{\nu}\left(\cos\theta\right)=-\frac{\pi}{2\nu^{\mu}}\left(% \frac{\theta}{\sin\theta}\right)^{1/2}\left(Y_{\mu}\left(\left(\nu+\tfrac{1}{2% }\right)\theta\right)+O\left(\frac{1}{\nu}\right)\mathrm{env}\mskip-2.0mu Y_{% \mu}\left(\left(\nu+\tfrac{1}{2}\right)\theta\right)\right),$
14.15.15 $\mathsf{P}^{-\mu}_{\nu}\left(x\right)=\beta\left(\frac{y-\alpha^{2}}{1-\alpha^% {2}-x^{2}}\right)^{1/4}\*\left(J_{\mu}\left(\left(\nu+\tfrac{1}{2}\right)y^{1/% 2}\right)+O\left(\frac{1}{\nu}\right)\mathrm{env}\mskip-2.0mu J_{\mu}\left(% \left(\nu+\tfrac{1}{2}\right)y^{1/2}\right)\right),$
Here we introduce the envelopes of the parabolic cylinder functions $U\left(-c,x\right)$, $\overline{U}\left(-c,x\right)$, which are defined in §12.2. For $U\left(-c,x\right)$ or $\overline{U}\left(-c,x\right)$, with $c$ and $x$ nonnegative, …
##### 6: 13.8 Asymptotic Approximations for Large Parameters
13.8.9 $M\left(a,b,x\right)=\Gamma\left(b\right)e^{\frac{1}{2}x}\left((\tfrac{1}{2}b-a% )x\right)^{\frac{1}{2}-\frac{1}{2}b}\*\left(J_{b-1}\left(\sqrt{2x(b-2a)}\right% )+\mathrm{env}\mskip-2.0mu J_{b-1}\left(\sqrt{2x(b-2a)}\right)O\left({\left|a% \right|}^{-\frac{1}{2}}\right)\right),$
13.8.10 $U\left(a,b,x\right)=\Gamma\left(\tfrac{1}{2}b-a+\tfrac{1}{2}\right)e^{\frac{1}% {2}x}x^{\frac{1}{2}-\frac{1}{2}b}\*\left(\cos\left(a\pi\right)J_{b-1}\left(% \sqrt{2x(b-2a)}\right)-\sin\left(a\pi\right)Y_{b-1}\left(\sqrt{2x(b-2a)}\right% )+\mathrm{env}\mskip-2.0mu Y_{b-1}\left(\sqrt{2x(b-2a)}\right)O\left({\left|a% \right|}^{-\frac{1}{2}}\right)\right),$
##### 7: 18.15 Asymptotic Approximations
18.15.19 $L^{(\alpha)}_{n}\left(\nu x\right)=\frac{e^{\frac{1}{2}\nu x}}{2^{\alpha}x^{% \frac{1}{2}\alpha+\frac{1}{4}}(1-x)^{\frac{1}{4}}}\left(\xi^{\frac{1}{2}}J_{% \alpha}\left(\nu\xi\right)\sum_{m=0}^{M-1}\frac{A_{m}(\xi)}{\nu^{2m}}+\xi^{-% \frac{1}{2}}J_{\alpha+1}\left(\nu\xi\right)\sum_{m=0}^{M-1}\frac{B_{m}(\xi)}{% \nu^{2m+1}}+\xi^{\frac{1}{2}}\mathrm{env}\mskip-2.0mu J_{\alpha}\left(\nu\xi% \right)O\left(\frac{1}{\nu^{2M-1}}\right)\right),$
Here $J_{\nu}\left(z\right)$ denotes the Bessel function10.2(ii)), $\mathrm{env}\mskip-2.0mu J_{\nu}\left(z\right)$ denotes its envelope2.8(iv)), and $\delta$ is again an arbitrary small positive constant. …
18.15.22 $L^{(\alpha)}_{n}\left(\nu x\right)=(-1)^{n}\frac{e^{\frac{1}{2}\nu x}}{2^{% \alpha-\frac{1}{2}}x^{\frac{1}{2}\alpha+\frac{1}{4}}}\*\left(\frac{\zeta}{x-1}% \right)^{\frac{1}{4}}\left(\frac{\mathrm{Ai}\left(\nu^{\frac{2}{3}}\zeta\right% )}{\nu^{\frac{1}{3}}}\sum_{m=0}^{M-1}\frac{E_{m}(\zeta)}{\nu^{2m}}+\frac{% \mathrm{Ai}'\left(\nu^{\frac{2}{3}}\zeta\right)}{\nu^{\frac{5}{3}}}\sum_{m=0}^% {M-1}\frac{F_{m}(\zeta)}{\nu^{2m}}+\mathrm{envAi}\left(\nu^{\frac{2}{3}}\zeta% \right)O\left(\frac{1}{\nu^{2M-\frac{2}{3}}}\right)\right),$
Here $\mathrm{Ai}$ denotes the Airy function9.2), $\mathrm{Ai}'$ denotes its derivative, and $\mathrm{envAi}$ denotes its envelope2.8(iii)). …
##### 8: Errata
• Equation (14.15.23)

Originally used $f(x)$ to represent both $U\left(-c,x\right)$ and $\overline{U}\left(-c,x\right)$. This has been replaced by two equations giving explicit definitions for the two envelope functions. Some slight changes in wording were needed to make this clear to readers.

• ##### 9: 12.2 Differential Equations
###### §12.2(i) Introduction
All solutions are entire functions of $z$ and entire functions of $a$ or $\nu$. …
###### §12.2(iii) Wronskians
Its importance is that when $a$ is negative and $|a|$ is large, $U\left(a,x\right)$ and $\overline{U}\left(a,x\right)$ asymptotically have the same envelope (modulus) and are $\tfrac{1}{2}\pi$ out of phase in the oscillatory interval $-2\sqrt{-a}. …