parabolic cylinder functions

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1: 12.14 The Function $W(a,x)$

§12.14 The Function $W(a,x)$

… ►This equation is important when $a$ and $z$ $(=x)$ are real, and we shall assume this to be the case. … ►

Bessel Functions

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Confluent Hypergeometric Functions

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§12.14(x) Modulus and Phase Functions

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2: 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 $D_{\nu }\left(z\right)$. …

3: 12.2 Differential Equations

§12.2 Differential Equations

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§12.2(i) Introduction

… ►Standard solutions are $U(a,\pm z)$, $V(a,\pm z)$, $\overline{U}(a,\pm x)$ (not complex conjugate), $U(-a,\pm \mathrm{i}z)$ for (12.2.2); $W(a,\pm x)$ for (12.2.3); $D_{\nu }\left(\pm z\right)$ for (12.2.4), where … ►

§12.2(iii) Wronskians

… ►When $z$ $(=x)$ is real the solution $\overline{U}(a,x)$ is defined by …

4: 14.15 Uniform Asymptotic Approximations

… ►Here we introduce the envelopes of the parabolic cylinder functions $U(-c,x)$, $\overline{U}(-c,x)$, which are defined in §12.2. For $U(-c,x)$ or $\overline{U}(-c,x)$, with $c$ and $x$ nonnegative, … ►

14.15.24 $${\mathsf{P}}_{\nu }^{-\mu }\left(x\right)=\frac{1}{{\left(\nu +\frac{1}{2}\right)}^{1/4}{2}^{(\nu +\mu )/2}\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +\frac{3}{4}\right)}{\left(\frac{{\zeta }^2-{\alpha }^2}{x^2-a^2}\right)}^{1/4}\left(U(\mu -\nu -\frac{1}{2},{\left(2\nu +1\right)}^{1/2}\zeta )+O\left({\nu }^{-2/3}\right)\mathrm{env}U(\mu -\nu -\frac{1}{2},{\left(2\nu +1\right)}^{1/2}\zeta )\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathsf{P}}_{\nu }^{\mu }\left(x\right)$: Ferrers function of the first kind, $\mathrm{env}U(c,x)$: envelope of parabolic cylinder function $U(c,x)$ , $U(a,z)$: parabolic cylinder function, $x$: real variable, $\mu$: general order, $\nu$: general degree, $a$ , $\zeta$ and $\alpha$

Permalink:

http://dlmf.nist.gov/14.15.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(v), §14.15 and Ch.14

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14.15.25 $${\mathsf{Q}}_{\nu }^{-\mu }\left(x\right)=\frac{\pi }{{\left(\nu +\frac{1}{2}\right)}^{1/4}{2}^{(\nu +\mu +2)/2}\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +\frac{3}{4}\right)}{\left(\frac{{\zeta }^2-{\alpha }^2}{x^2-a^2}\right)}^{1/4}\left(\overline{U}(\mu -\nu -\frac{1}{2},{\left(2\nu +1\right)}^{1/2}\zeta )+O\left({\nu }^{-2/3}\right)\mathrm{env}\overline{U}(\mu -\nu -\frac{1}{2},{\left(2\nu +1\right)}^{1/2}\zeta )\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathsf{Q}}_{\nu }^{\mu }\left(x\right)$: Ferrers function of the second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{env}\overline{U}(c,x)$: envelope of parabolic cylinder function $\overline{U}(c,x)$ , $\overline{U}(a,x)$: parabolic cylinder function, $x$: real variable, $\mu$: general order, $\nu$: general degree, $a$ , $\zeta$ and $\alpha$

Permalink:

http://dlmf.nist.gov/14.15.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(v), §14.15 and Ch.14

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14.15.30 $${\mathsf{P}}_{\nu }^{-\mu }\left(x\right)=\frac{1}{{\left(\nu +\frac{1}{2}\right)}^{1/4}{2}^{(\nu +\mu )/2}\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +\frac{3}{4}\right)}{\left(\frac{{\zeta }^2+{\alpha }^2}{x^2+a^2}\right)}^{1/4}U(\mu -\nu -\frac{1}{2},{\left(2\nu +1\right)}^{1/2}\zeta )\left(1+O\left({\nu }^{-1}\ln \nu \right)\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\mathsf{P}}_{\nu }^{\mu }\left(x\right)$: Ferrers function of the first kind, $\ln z$: principal branch of logarithm function, $U(a,z)$: parabolic cylinder function, $x$: real variable, $\mu$: general order, $\nu$: general degree, $a$ , $\zeta$ and $\alpha$

Referenced by:

§14.15(v)

Permalink:

http://dlmf.nist.gov/14.15.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §14.15(v), §14.15 and Ch.14

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5: 12.16 Mathematical Applications

§12.16 Mathematical Applications

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

6: 12.15 Generalized Parabolic Cylinder Functions

§12.15 Generalized Parabolic Cylinder Functions

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7: 12 Parabolic Cylinder Functions

Chapter 12 Parabolic Cylinder Functions

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8: 12.6 Continued Fraction

§12.6 Continued Fraction

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

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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

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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