parabolic cylinder functions

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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: $U(a,z)$, $V(a,z)$, $\overline{U}(a,z)$, and $W(a,z)$. …An older notation, due to Whittaker (1902), for $U(a,z)$ is $D_{\nu }\left(z\right)$. …

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

§12.14 The Function $W(a,x)$

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§12.14(iii) Graphs

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Bessel Functions

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Confluent Hypergeometric Functions

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§12.14(x) Modulus and Phase 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: 12.15 Generalized Parabolic Cylinder Functions

§12.15 Generalized Parabolic Cylinder Functions

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5: 12 Parabolic Cylinder Functions

Chapter 12 Parabolic Cylinder Functions

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6: 12.6 Continued Fraction

§12.6 Continued Fraction

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7: 12.20 Approximations

§12.20 Approximations

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8: 12.7 Relations to Other Functions

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§12.7(i) Hermite Polynomials

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12.7.1 $$U(-\frac{1}{2},z)=D_0\left(z\right)={\mathrm{e}}^{-\frac{1}{4}{z}^2},$$

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Symbols:

$D_{\nu }\left(z\right)$: parabolic cylinder function, $\mathrm{e}$: base of natural logarithm, $U(a,z)$: parabolic cylinder function and $z$: complex variable

Permalink:

http://dlmf.nist.gov/12.7.E1

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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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9: 12.18 Methods of Computation

§12.18 Methods of Computation

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10: 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. … ►Lastly, parabolic cylinder functions arise in the description of ultra cold atoms in harmonic trapping potentials; see Busch et al. (1998) and Edwards et al. (1999).