# parabolic cylinder functions

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##### 1: 12.14 The Function $W\left(a,x\right)$
###### §12.14 The Function$W\left(a,x\right)$
This equation is important when $a$ and $z$ $(=x)$ are real, and we shall assume this to be the case. …
##### 2: 12.1 Special Notation
Unless otherwise noted, primes indicate derivatives with respect to the variable, and fractional powers take their principal values. The main functions treated in this chapter are the parabolic cylinder functions (PCFs), also known as Weber parabolic cylinder functions: $U\left(a,z\right)$, $V\left(a,z\right)$, $\overline{U}\left(a,z\right)$, and $W\left(a,z\right)$. …An older notation, due to Whittaker (1902), for $U\left(a,z\right)$ is $D_{\nu}\left(z\right)$. …
##### 3: 12.2 Differential Equations
###### §12.2(i) Introduction
Standard solutions are $U\left(a,\pm z\right)$, $V\left(a,\pm z\right)$, $\overline{U}\left(a,\pm x\right)$ (not complex conjugate), $U\left(-a,\pm iz\right)$ for (12.2.2); $W\left(a,\pm x\right)$ for (12.2.3); $D_{\nu}\left(\pm z\right)$ for (12.2.4), where …
###### §12.2(iii) Wronskians
When $z$ $(=x)$ is real the solution $\overline{U}\left(a,x\right)$ is defined by …
##### 4: 14.15 Uniform Asymptotic Approximations
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, …
14.15.24 $\mathsf{P}^{-\mu}_{\nu}\left(x\right)=\frac{1}{\left(\nu+\frac{1}{2}\right)^{1% /4}2^{(\nu+\mu)/2}\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{4}\right)% }\left(\frac{\zeta^{2}-\alpha^{2}}{x^{2}-a^{2}}\right)^{1/4}\*\left(U\left(\mu% -\nu-\tfrac{1}{2},\left(2\nu+1\right)^{1/2}\zeta\right)+O\left(\nu^{-2/3}% \right)\mathrm{env}\mskip-1.0mu U\left(\mu-\nu-\tfrac{1}{2},\left(2\nu+1\right% )^{1/2}\zeta\right)\right),$
14.15.25 $\mathsf{Q}^{-\mu}_{\nu}\left(x\right)=\frac{\pi}{\left(\nu+\frac{1}{2}\right)^% {1/4}2^{(\nu+\mu+2)/2}\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{4}% \right)}\*\left(\frac{\zeta^{2}-\alpha^{2}}{x^{2}-a^{2}}\right)^{1/4}\*\left(% \overline{U}\left(\mu-\nu-\tfrac{1}{2},\left(2\nu+1\right)^{1/2}\zeta\right)+O% \left(\nu^{-2/3}\right)\mathrm{env}\mskip-1.0mu \overline{U}\left(\mu-\nu-% \tfrac{1}{2},\left(2\nu+1\right)^{1/2}\zeta\right)\right),$
14.15.30 $\mathsf{P}^{-\mu}_{\nu}\left(x\right)=\frac{1}{\left(\nu+\frac{1}{2}\right)^{1% /4}2^{(\nu+\mu)/2}\Gamma\left(\frac{1}{2}\nu+\frac{1}{2}\mu+\frac{3}{4}\right)% }\left(\frac{\zeta^{2}+\alpha^{2}}{x^{2}+a^{2}}\right)^{1/4}\*U\left(\mu-\nu-% \tfrac{1}{2},\left(2\nu+1\right)^{1/2}\zeta\right)\left(1+O\left(\nu^{-1}\ln% \nu\right)\right),$
##### 5: 12.16 Mathematical Applications
###### §12.16 Mathematical Applications
For examples see §§13.20(iii), 13.20(iv), 14.15(v), and 14.26. …
##### 10: 12.7 Relations to Other Functions
###### §12.7(i) Hermite Polynomials
12.7.1 $U\left(-\tfrac{1}{2},z\right)=D_{0}\left(z\right)=e^{-\frac{1}{4}z^{2}},$