# parabolic cylinder

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##### 1: 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: 12.16 Mathematical Applications
###### §12.16 Mathematical Applications
For examples see §§13.20(iii), 13.20(iv), 14.15(v), and 14.26. …
##### 5: 12.17 Physical Applications
###### §12.17 Physical Applications
By using instead coordinates of the parabolic cylinder $\xi,\eta,\zeta$, defined by … Dean (1966) describes the role of PCFs in quantum mechanical systems closely related to the one-dimensional harmonic oscillator. Problems on high-frequency scattering in homogeneous media by parabolic cylinders lead to asymptotic methods for integrals involving PCFs. …
##### 10: 12.7 Relations to Other Functions
###### §12.7(i) Hermite Polynomials
12.7.1 $U\left(-\tfrac{1}{2},z\right)=D_{0}\left(z\right)=e^{-\frac{1}{4}z^{2}},$