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)

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

. …

2: 12.14 The Function $W\left(a,x\right)$

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

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§12.14(iii) Graphs

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§12.14(iv) Connection Formula

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Bessel Functions

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3: 12.16 Mathematical Applications

§12.16 Mathematical Applications

… ►For examples see §§13.20(iii), 13.20(iv), 14.15(v), and 14.26. … ► … ►

4: 12.6 Continued Fraction

§12.6 Continued Fraction

…

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. …

6: 12 Parabolic Cylinder Functions

Chapter 12 Parabolic Cylinder Functions

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7: 12.18 Methods of Computation

§12.18 Methods of Computation

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8: 12.15 Generalized Parabolic Cylinder Functions

§12.15 Generalized Parabolic Cylinder Functions

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9: 12.20 Approximations

§12.20 Approximations

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10: 12.7 Relations to Other Functions

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§12.7(i) Hermite Polynomials

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12.7.1

U\left(-\tfrac{1}{2},z\right)=D_{0}\left(z\right)=e^{-\frac{1}{4}z^{2}},

ⓘ

Symbols:: $D_{\NVar{\nu}}\left(\NVar{z}\right)$ : parabolic cylinder function, $\mathrm{e}$ : base of natural logarithm, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/12.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §12.7(i), §12.7 and Ch.12

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

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§12.7(iii) Modified Bessel Functions

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§12.7(iv) Confluent Hypergeometric Functions

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