… ►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)

D_{\nu}\left(z\right)

. The notations are related by

U\left(a,z\right)=D_{-a-\frac{1}{2}}\left(z\right)

. …

10: 18.15 Asymptotic Approximations

… ►With

\mu=\sqrt{2n+1}

the expansions in Chapter 12 are for the parabolic cylinder function

U\left(-\tfrac{1}{2}\mu^{2},\mu t\sqrt{2}\right)

, which is related to the Hermite polynomials via …

relation to parabolic cylinder functions

1—10 of 31 matching pages

1: 12.7 Relations to Other Functions

§12.7(i) Hermite Polynomials

§12.7(ii) Error Functions, Dawson’s Integral, and Probability Function

§12.7(iii) Modified Bessel Functions

§12.7(iv) Confluent Hypergeometric Functions

2: 10.39 Relations to Other Functions

Parabolic Cylinder Functions

3: 10.16 Relations to Other Functions

Parabolic Cylinder Functions

4: 13.18 Relations to Other Functions

§13.18(iv) Parabolic Cylinder Functions

5: 13.6 Relations to Other Functions

§13.6(iv) Parabolic Cylinder Functions

6: 12.14 The Function W ⁡ ( a , x )

Bessel Functions

Confluent Hypergeometric Functions

7: 7.18 Repeated Integrals of the Complementary Error Function

Parabolic Cylinder Functions

Probability Functions

8: 12.17 Physical Applications

9: 12.1 Special Notation

10: 18.15 Asymptotic Approximations

6: 12.14 The Function $W\left(a,x\right)$