12 Parabolic Cylinder Functions Properties 12.4 Power-Series Expansions 12.6 Continued Fraction

§12.5 Integral Representations

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Permalink:: http://dlmf.nist.gov/12.5
See also:: Annotations for Ch.12

§12.5(i) Integrals Along the Real Line

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Keywords:: along the real line, integral representations, parabolic cylinder functions
Notes:: See Miller (1955, pp. 19, 25) and Whittaker (1902). For (12.5.4) combine (12.2.18) and (12.5.1).
Referenced by:: §12.5(ii)
Permalink:: http://dlmf.nist.gov/12.5.i
See also:: Annotations for §12.5 and Ch.12

12.5.1		$U\left(a,z\right)=\frac{e^{-\frac{1}{4}z^{2}}}{\Gamma\left(\frac{1}{2}+a\right% )}\int_{0}^{\infty}t^{a-\frac{1}{2}}e^{-\frac{1}{2}t^{2}-zt}\,\mathrm{d}t,$
$\Re a>-\tfrac{1}{2}$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma 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, $z$ : complex variable and $a$ : real or complex parameter A&S Ref: 19.5.3 (modification of) Referenced by: §12.5(i), §12.8(ii), §36.2(ii) Permalink: http://dlmf.nist.gov/12.5.E1 Encodings: TeX, pMML, png See also: Annotations for §12.5(i), §12.5 and Ch.12

12.5.2		$U\left(a,z\right)=\frac{ze^{-\frac{1}{4}z^{2}}}{\Gamma\left(\frac{1}{4}+\frac{% 1}{2}a\right)}\*\int_{0}^{\infty}t^{\frac{1}{2}a-\frac{3}{4}}e^{-t}\left(z^{2}% +2t\right)^{-\frac{1}{2}a-\frac{3}{4}}\,\mathrm{d}t,$
$\|\operatorname{ph}z\|<\tfrac{1}{2}\pi$ , $\Re a>-\tfrac{1}{2}$ ,
ⓘ 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, $\int$ : integral, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $\operatorname{ph}$ : phase, $\Re$ : real part, $z$ : complex variable and $a$ : real or complex parameter A&S Ref: 19.5.11 (modification of) Permalink: http://dlmf.nist.gov/12.5.E2 Encodings: TeX, pMML, png See also: Annotations for §12.5(i), §12.5 and Ch.12

12.5.3		$U\left(a,z\right)=\frac{e^{-\frac{1}{4}z^{2}}}{\Gamma\left(\frac{3}{4}+\frac{1% }{2}a\right)}\*\int_{0}^{\infty}t^{\frac{1}{2}a-\frac{1}{4}}e^{-t}\left(z^{2}+% 2t\right)^{-\frac{1}{2}a-\frac{1}{4}}\,\mathrm{d}t,$
$\|\operatorname{ph}z\|<\tfrac{1}{2}\pi$ , $\Re a>-\tfrac{3}{2}$ ,
ⓘ 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, $\int$ : integral, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $\operatorname{ph}$ : phase, $\Re$ : real part, $z$ : complex variable and $a$ : real or complex parameter A&S Ref: 19.5.12 (modification of) Permalink: http://dlmf.nist.gov/12.5.E3 Encodings: TeX, pMML, png See also: Annotations for §12.5(i), §12.5 and Ch.12

12.5.4		$U\left(a,z\right)=\sqrt{\frac{2}{\pi}}e^{\frac{1}{4}z^{2}}\*\int_{0}^{\infty}t% ^{-a-\frac{1}{2}}e^{-\frac{1}{2}t^{2}}\cos\left(zt+\left(\tfrac{1}{2}a+\tfrac{% 1}{4}\right)\pi\right)\,\mathrm{d}t,$
$\Re a<\tfrac{1}{2}$ .
ⓘ Symbols: $\pi$ : 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, $z$ : complex variable and $a$ : real or complex parameter Referenced by: §12.5(i) Permalink: http://dlmf.nist.gov/12.5.E4 Encodings: TeX, pMML, png See also: Annotations for §12.5(i), §12.5 and Ch.12

§12.5(ii) Contour Integrals

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Keywords:: contour integrals, integral representations, parabolic cylinder functions
Referenced by:: §12.14(vi)
Permalink:: http://dlmf.nist.gov/12.5.ii
Clarification (effective with 1.2.1):: Just below (12.5.5), we added “For the particular loop contour, see Figure 5.9.1.”
See also:: Annotations for §12.5 and Ch.12

The following integrals correspond to those of §12.5(i).

12.5.5		$U\left(a,z\right)=\frac{\Gamma\left(\frac{1}{2}-a\right)}{2\pi i}e^{-\frac{1}{% 4}z^{2}}\int_{-\infty}^{(0+)}e^{zt-\frac{1}{2}t^{2}}t^{a-\frac{1}{2}}\,\mathrm% {d}t,$
$a\neq\frac{1}{2},\frac{3}{2},\frac{5}{2},\dots$ , $-\pi<\operatorname{ph}t<\pi$ .
ⓘ 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, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $\operatorname{ph}$ : phase, $z$ : complex variable and $a$ : real or complex parameter A&S Ref: 19.5.1 (modification of) Referenced by: §12.5(ii), Erratum (V1.2.1) for Figure 5.9.1 Permalink: http://dlmf.nist.gov/12.5.E5 Encodings: TeX, pMML, png See also: Annotations for §12.5(ii), §12.5 and Ch.12

For the particular loop contour, see Figure 5.9.1.

Restrictions on $a$ are not needed in the following two representations:

12.5.6		$U\left(a,z\right)=\frac{e^{\frac{1}{4}z^{2}}}{i\sqrt{2\pi}}\int_{c-i\infty}^{c% +i\infty}e^{-zt+\frac{1}{2}t^{2}}t^{-a-\frac{1}{2}}\,\mathrm{d}t,$
$-\tfrac{1}{2}\pi<\operatorname{ph}t<\tfrac{1}{2}\pi$ , $c>0$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $\operatorname{ph}$ : phase, $z$ : complex variable and $a$ : real or complex parameter A&S Ref: 19.5.4 (modification of) Referenced by: §12.8(ii) Permalink: http://dlmf.nist.gov/12.5.E6 Encodings: TeX, pMML, png See also: Annotations for §12.5(ii), §12.5 and Ch.12

12.5.7		$V\left(a,z\right)=\frac{e^{-\frac{1}{4}z^{2}}}{2\pi}\*\left(\int_{-ic-\infty}^% {-ic+\infty}+\int_{ic-\infty}^{ic+\infty}\right)e^{zt-\frac{1}{2}t^{2}}t^{a-% \frac{1}{2}}\,\mathrm{d}t,$
$-\pi<\operatorname{ph}{t}<\pi$ , $c>0$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $V\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $\operatorname{ph}$ : phase, $z$ : complex variable and $a$ : real or complex parameter Referenced by: §12.8(ii) Permalink: http://dlmf.nist.gov/12.5.E7 Encodings: TeX, pMML, png See also: Annotations for §12.5(ii), §12.5 and Ch.12

For proofs and further results see Miller (1955, §4) and Whittaker (1902).

§12.5(iii) Mellin–Barnes Integrals

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Keywords:: Mellin–Barnes type, integral representations, parabolic cylinder functions
Notes:: See Miller (1955, p. 26). In this reference the conditions on $a$ given in Eqs. (12.5.8) and (12.5.9) are missing. These conditions are needed to ensure that in each integrand no poles of the two gamma functions coincide.
Permalink:: http://dlmf.nist.gov/12.5.iii
See also:: Annotations for §12.5 and Ch.12

12.5.8	$\displaystyle U\left(a,z\right)$	$\displaystyle=\frac{e^{-\frac{1}{4}z^{2}}z^{-a-\frac{1}{2}}}{2\pi i\Gamma\left% (\frac{1}{2}+a\right)}\*\int_{-i\infty}^{i\infty}\Gamma\left(t\right)\Gamma% \left(\tfrac{1}{2}+a-2t\right)2^{t}z^{2t}\,\mathrm{d}t,$
$a\neq-\frac{1}{2},-\frac{3}{2},-\frac{5}{2},\dots$ , $\|\operatorname{ph}z\|<\tfrac{3}{4}\pi$ ,
ⓘ 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, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $U\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $\operatorname{ph}$ : phase, $z$ : complex variable and $a$ : real or complex parameter Keywords: Mellin transform A&S Ref: 19.5.13 (modification and correction of) Referenced by: §12.5(iii) Permalink: http://dlmf.nist.gov/12.5.E8 Encodings: TeX, pMML, png See also: Annotations for §12.5(iii), §12.5 and Ch.12
where the contour separates the poles of $\Gamma\left(t\right)$ from those of $\Gamma\left(\tfrac{1}{2}+a-2t\right)$ .
12.5.9	$\displaystyle V\left(a,z\right)$	$\displaystyle=\sqrt{\frac{2}{\pi}}\frac{e^{\frac{1}{4}z^{2}}z^{a-\frac{1}{2}}}% {2\pi i\Gamma\left(\frac{1}{2}-a\right)}\*\int_{-i\infty}^{i\infty}\Gamma\left% (t\right)\Gamma\left(\tfrac{1}{2}-a-2t\right)2^{t}z^{2t}\cos\left(\pi t\right)% \,\mathrm{d}t,$
$a\neq\frac{1}{2},\frac{3}{2},\frac{5}{2},\dots$ , $\|\operatorname{ph}z\|<\tfrac{1}{4}\pi$ ,
ⓘ 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, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $V\left(\NVar{a},\NVar{z}\right)$ : parabolic cylinder function, $\operatorname{ph}$ : phase, $z$ : complex variable and $a$ : real or complex parameter Keywords: Mellin transform A&S Ref: 19.5.13 (modification and correction of) Referenced by: §12.5(iii), §12.8(ii) Permalink: http://dlmf.nist.gov/12.5.E9 Encodings: TeX, pMML, png See also: Annotations for §12.5(iii), §12.5 and Ch.12

where the contour separates the poles of $\Gamma\left(t\right)$ from those of $\Gamma\left(\tfrac{1}{2}-a-2t\right)$ .

§12.5(iv) Compendia

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Keywords:: compendia, integral representations, parabolic cylinder functions
Permalink:: http://dlmf.nist.gov/12.5.iv
See also:: Annotations for §12.5 and Ch.12

For further collections of integral representations see Apelblat (1983, pp. 427-436), Erdélyi et al. (1953b, v. 2, pp. 119–120), Erdélyi et al. (1954a, pp. 289–291 and 362), Gradshteyn and Ryzhik (2015, §§9.24–9.25), Magnus et al. (1966, pp. 328–330), Oberhettinger (1974, pp. 251–252), and Oberhettinger and Badii (1973, pp. 378–384).

12.4 Power-Series Expansions 12.6 Continued Fraction

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§12.5 Integral Representations

Contents

§12.5(i) Integrals Along the Real Line

§12.5(ii) Contour Integrals

§12.5(iii) Mellin–Barnes Integrals

§12.5(iv) Compendia

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