§12.5 Integral Representations

12.5.1		$$U(a,z)=\frac{{\mathrm{e}}^{-\frac{1}{4}{z}^2}}{\mathrm{\Gamma }\left(\frac{1}{2}+a\right)}{\int }_0^{\mathrm{\infty }}{t}^{a-\frac{1}{2}}{\mathrm{e}}^{-\frac{1}{2}{t}^2-zt}dt,$$
$\mathrm{\Re }a>-\frac{1}{2}$ ,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\mathrm{\Re }$: 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		$$U(a,z)=\frac{z{\mathrm{e}}^{-\frac{1}{4}{z}^2}}{\mathrm{\Gamma }\left(\frac{1}{4}+\frac{1}{2}a\right)}{\int }_0^{\mathrm{\infty }}{t}^{\frac{1}{2}a-\frac{3}{4}}{\mathrm{e}}^{-t}{\left(z^2+2t\right)}^{-\frac{1}{2}a-\frac{3}{4}}dt,$$
, $\mathrm{\Re }a>-\frac{1}{2}$ ,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $\mathrm{\Re }$: 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		$$U(a,z)=\frac{{\mathrm{e}}^{-\frac{1}{4}{z}^2}}{\mathrm{\Gamma }\left(\frac{3}{4}+\frac{1}{2}a\right)}{\int }_0^{\mathrm{\infty }}{t}^{\frac{1}{2}a-\frac{1}{4}}{\mathrm{e}}^{-t}{\left(z^2+2t\right)}^{-\frac{1}{2}a-\frac{1}{4}}dt,$$
, $\mathrm{\Re }a>-\frac{3}{2}$ ,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $\mathrm{\Re }$: 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		$$U(a,z)=\sqrt{\frac{2}{\pi }}{\mathrm{e}}^{\frac{1}{4}{z}^2}{\int }_0^{\mathrm{\infty }}{t}^{-a-\frac{1}{2}}{\mathrm{e}}^{-\frac{1}{2}{t}^2}\cos \left(zt+\left(\frac{1}{2}a+\frac{1}{4}\right)\pi \right)dt,$$
.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\mathrm{\Re }$: 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

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

12.5.5		$$U(a,z)=\frac{\mathrm{\Gamma }\left(\frac{1}{2}-a\right)}{2\pi \mathrm{i}}{\mathrm{e}}^{-\frac{1}{4}{z}^2}{\int }_{-\mathrm{\infty }}^{(0+)}{\mathrm{e}}^{zt-\frac{1}{2}{t}^2}{t}^{a-\frac{1}{2}}dt,$$
$a\ne \frac{1}{2},\frac{3}{2},\frac{5}{2},\mathrm{\dots }$, .
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable and $a$: real or complex parameter A&S Ref: 19.5.1 (modification of) Permalink: http://dlmf.nist.gov/12.5.E5 Encodings: TeX, pMML, png See also: Annotations for §12.5(ii), §12.5 and Ch.12

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

12.5.6		$$U(a,z)=\frac{{\mathrm{e}}^{\frac{1}{4}{z}^2}}{\mathrm{i}\sqrt{2\pi }}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}{\mathrm{e}}^{-zt+\frac{1}{2}{t}^2}{t}^{-a-\frac{1}{2}}dt,$$
, $c>0$ ,
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\mathrm{ph}$: 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		$$V(a,z)=\frac{{\mathrm{e}}^{-\frac{1}{4}{z}^2}}{2\pi }\left({\int }_{-\mathrm{i}c-\mathrm{\infty }}^{-\mathrm{i}c+\mathrm{\infty }}+{\int }_{\mathrm{i}c-\mathrm{\infty }}^{\mathrm{i}c+\mathrm{\infty }}\right){\mathrm{e}}^{zt-\frac{1}{2}{t}^2}{t}^{a-\frac{1}{2}}dt,$$
, $c>0$.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $V(a,z)$: parabolic cylinder function, $\mathrm{ph}$: 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).

12.5.8		$U(a,z)$	$={\displaystyle \frac{{\mathrm{e}}^{-\frac{1}{4}{z}^2}{z}^{-a-\frac{1}{2}}}{2\pi \mathrm{i}\mathrm{\Gamma }\left(\frac{1}{2}+a\right)}}{\displaystyle {\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}}\mathrm{\Gamma }\left(t\right)\mathrm{\Gamma }\left(\frac{1}{2}+a-2t\right)2^t{z}^{2t}dt,$
$a\ne -\frac{1}{2},-\frac{3}{2},-\frac{5}{2},\mathrm{\dots }$, ,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\mathrm{ph}$: 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 $\mathrm{\Gamma }\left(t\right)$ from those of $\mathrm{\Gamma }\left(\frac{1}{2}+a-2t\right)$.
12.5.9		$V(a,z)$	$=\sqrt{{\displaystyle \frac{2}{\pi }}}{\displaystyle \frac{{\mathrm{e}}^{\frac{1}{4}{z}^2}{z}^{a-\frac{1}{2}}}{2\pi \mathrm{i}\mathrm{\Gamma }\left(\frac{1}{2}-a\right)}}{\displaystyle {\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}}\mathrm{\Gamma }\left(t\right)\mathrm{\Gamma }\left(\frac{1}{2}-a-2t\right)2^t{z}^{2t}\cos \left(\pi t\right)dt,$
$a\ne \frac{1}{2},\frac{3}{2},\frac{5}{2},\mathrm{\dots }$, ,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $V(a,z)$: parabolic cylinder function, $\mathrm{ph}$: 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 $\mathrm{\Gamma }\left(t\right)$ from those of $\mathrm{\Gamma }\left(\frac{1}{2}-a-2t\right)$.

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 (2000, §§9.24–9.25), Magnus et al. (1966, pp. 328–330), Oberhettinger (1974, pp. 251–252), and Oberhettinger and Badii (1973, pp. 378–384).