(For other notation see Notation for the Special Functions.)

$x,y$ | real variables. |
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$z$ | complex variable. |

$n,s$ | nonnegative integers. |

$a,\nu $ | real or complex parameters. |

$\delta $ | arbitrary small positive constant. |

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 $W\left(a,z\right)$. These notations are due to Miller (1952, 1955). An older notation, due to Whittaker (1902), for $U\left(a,z\right)$ is ${D}_{\nu}\left(z\right)$. The notations are related by $U\left(a,z\right)={D}_{-a-\frac{1}{2}}\left(z\right)$. Whittaker’s notation ${D}_{\nu}\left(z\right)$ is useful when $\nu $ is a nonnegative integer (Hermite polynomial case).