12 Parabolic Cylinder Functions Properties 12.11 Zeros 12.13 Sums

§12.12 Integrals

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Keywords:: integrals, parabolic cylinder functions
Notes:: See Erdélyi et al. (1953b, v. 2, Chapter 8). For (12.12.4) see Durand (1975).
Permalink:: http://dlmf.nist.gov/12.12
See also:: Annotations for Ch.12

12.12.1		$\int_{0}^{\infty}e^{-\frac{1}{4}t^{2}}t^{\mu-1}U\left(a,t\right)\,\mathrm{d}t=% \frac{\sqrt{\pi}2^{-\frac{1}{2}(\mu+a+\frac{1}{2})}\Gamma\left(\mu\right)}{% \Gamma\left(\frac{1}{2}(\mu+a+\frac{3}{2})\right)},$
$\Re\mu>0$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $\Re$ : real part and $a$ : real or complex parameter Permalink: http://dlmf.nist.gov/12.12.E1 Encodings: TeX, pMML, png See also: Annotations for §12.12 and Ch.12

12.12.2		$\int_{0}^{\infty}e^{-\frac{3}{4}t^{2}}t^{-a-\frac{3}{2}}U\left(a,t\right)\,% \mathrm{d}t=2^{\frac{1}{4}+\frac{1}{2}a}\Gamma\left(-a-\tfrac{1}{2}\right)\cos% \left((\tfrac{1}{4}a+\tfrac{1}{8})\pi\right),$
$\Re a<-\tfrac{1}{2}$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $\Re$ : real part and $a$ : real or complex parameter Permalink: http://dlmf.nist.gov/12.12.E2 Encodings: TeX, pMML, png See also: Annotations for §12.12 and Ch.12

12.12.3		$\int_{0}^{\infty}e^{-\frac{1}{4}t^{2}}t^{-a-\frac{1}{2}}(x^{2}+t^{2})^{-1}U% \left(a,t\right)\,\mathrm{d}t=\sqrt{\pi/2}\Gamma\left(\tfrac{1}{2}-a\right)x^{% -a-\frac{3}{2}}e^{\frac{1}{4}x^{2}}U\left(-a,x\right),$
$\Re a<\tfrac{1}{2},x>0$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $\Re$ : real part, $x$ : real variable and $a$ : real or complex parameter Permalink: http://dlmf.nist.gov/12.12.E3 Encodings: TeX, pMML, png See also: Annotations for §12.12 and Ch.12

Nicholson-type Integral

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Keywords:: Nicholson-type, Nicholson-type integral, compendia, integrals, parabolic cylinder functions
See also:: Annotations for §12.12 and Ch.12

12.12.4		$(U\left(a,z\right))^{2}+(\overline{U}\left(a,z\right))^{2}=\frac{2^{\frac{3}{2% }}}{\pi}\Gamma\left(\tfrac{1}{2}-a\right)\int_{0}^{\infty}\frac{e^{2at+\frac{1% }{2}z^{2}\tanh t}}{\sqrt{\sinh\left(2t\right)}}\,\mathrm{d}t,$
$\Re a<\tfrac{1}{2}$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\sinh\NVar{z}$ : hyperbolic sine function, $\tanh\NVar{z}$ : hyperbolic tangent function, $\int$ : integral, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $\overline{U}\left(\NVar{a},\NVar{x}\right)$ : parabolic cylinder function, $\Re$ : real part, $z$ : complex variable and $a$ : real or complex parameter Notes: See Durand (1975). Referenced by: §12.12 Permalink: http://dlmf.nist.gov/12.12.E4 Encodings: TeX, pMML, png See also: Annotations for §12.12, §12.12 and Ch.12

When $z$ $(=x)$ is real the left-hand side equals $(F(a,x))^{2}$ ; compare (12.2.22).

For further integrals see §§13.10, 13.23, and use (12.7.14).

For compendia of integrals see Erdélyi et al. (1953b, v. 2, pp. 121–122), Erdélyi et al. (1954a, b, v. 1, pp. 60–61, 115, 210–211, and 336; v. 2, pp. 76–80, 115, 151, 171, and 395–398), Gradshteyn and Ryzhik (2015, §7.7), Magnus et al. (1966, pp. 330–331), Marichev (1983, pp. 190–191), Oberhettinger (1974, pp. 144–145), Oberhettinger (1990, pp. 106–108 and 192), Oberhettinger and Badii (1973, pp. 181–185), Prudnikov et al. (1986b, pp. 36–37, 155–168, 243–246, 289–290, 327–328, 419–420, and 619), Prudnikov et al. (1992a, §3.11), and Prudnikov et al. (1992b, §3.11).

§12.12 Integrals

Nicholson-type Integral

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