§12.5 Integral Representations

Permalink:: http://dlmf.nist.gov/12.5

§12.5(i) Integrals Along the Real Line
§12.5(ii) Contour Integrals
§12.5(iii) Mellin–Barnes Integrals
§12.5(iv) Compendia

§12.5(i) Integrals Along the Real Line

Notes:: See Miller (1955, pp. 19, 25) and Whittaker (1902). For (12.5.4) combine (12.2.18) and (12.5.1).
Keywords:: parabolic cylinder functions
Referenced by:: §12.5(ii)
Permalink:: http://dlmf.nist.gov/12.5.i

12.5.1 $\mathop{U\/}\nolimits\!\left(a,z\right)=\frac{e^{{-\frac{1}{4}z^{2}}}}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+a\right)}\int _{0}^{\infty}t^{{a-\frac{1}{2}}}e^{{-\frac{1}{2}t^{2}-zt}}dt,$ $\realpart{a}>-\tfrac{1}{2}$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\realpart{}$ : 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
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12.5.2 $\mathop{U\/}\nolimits\!\left(a,z\right)=\frac{ze^{{-\frac{1}{4}z^{2}}}}{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{4}+\frac{1}{2}a\right)}\*\int _{0}^{\infty}t^{{\frac{1}{2}a-\frac{3}{4}}}e^{{-t}}\left(z^{2}+2t\right)^{{-\frac{1}{2}a-\frac{3}{4}}}dt,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{2}\pi$ , $\realpart{a}>-\tfrac{1}{2}$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\realpart{}$ : 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

12.5.3 $\mathop{U\/}\nolimits\!\left(a,z\right)=\frac{e^{{-\frac{1}{4}z^{2}}}}{\mathop{\Gamma\/}\nolimits\!\left(\frac{3}{4}+\frac{1}{2}a\right)}\*\int _{0}^{\infty}t^{{\frac{1}{2}a-\frac{1}{4}}}e^{{-t}}\left(z^{2}+2t\right)^{{-\frac{1}{2}a-\frac{1}{4}}}dt,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{2}\pi$ , $\realpart{a}>-\tfrac{3}{2}$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\realpart{}$ : 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

12.5.4 $\mathop{U\/}\nolimits\!\left(a,z\right)=\sqrt{\frac{2}{\pi}}e^{{\frac{1}{4}z^{2}}}\*\int _{0}^{\infty}t^{{-a-\frac{1}{2}}}e^{{-\frac{1}{2}t^{2}}}\mathop{\cos\/}\nolimits\!\left(zt+\left(\tfrac{1}{2}a+\tfrac{1}{4}\right)\pi\right)dt,$ $\realpart{a}<\tfrac{1}{2}$ .

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\realpart{}$ : real part, $z$ : complex variable and $a$ : real or complex parameter
Referenced by:: §12.5(i)
Permalink:: http://dlmf.nist.gov/12.5.E4
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§12.5(ii) Contour Integrals

Keywords:: parabolic cylinder functions
Referenced by:: §12.14(vi)
Permalink:: http://dlmf.nist.gov/12.5.ii

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

12.5.5 $\mathop{U\/}\nolimits\!\left(a,z\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-a\right)}{2\pi i}e^{{-\frac{1}{4}z^{2}}}\int _{{-\infty}}^{{(0+)}}e^{{zt-\frac{1}{2}t^{2}}}t^{{a-\frac{1}{2}}}dt,$ $a\neq\frac{1}{2},\frac{3}{2},\frac{5}{2},\dots$ , $-\pi<\mathop{\mathrm{ph}\/}\nolimits t<\pi$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\mathop{\mathrm{ph}\/}\nolimits$ : 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

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

12.5.6 $\mathop{U\/}\nolimits\!\left(a,z\right)=\frac{e^{{\frac{1}{4}z^{2}}}}{i\sqrt{2\pi}}\int _{{c-i\infty}}^{{c+i\infty}}e^{{-zt+\frac{1}{2}t^{2}}}t^{{-a-\frac{1}{2}}}dt,$ $-\tfrac{1}{2}\pi<\mathop{\mathrm{ph}\/}\nolimits t<\tfrac{1}{2}\pi$ , $c>0$ ,

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\mathop{\mathrm{ph}\/}\nolimits$ : 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

12.5.7 $\mathop{V\/}\nolimits\!\left(a,z\right)=\frac{e^{{-\frac{1}{4}z^{2}}}}{2\pi}\*\left(\int _{{-ic-\infty}}^{{-ic+\infty}}+\int _{{ic-\infty}}^{{ic+\infty}}\right)e^{{zt-\frac{1}{2}t^{2}}}t^{{a-\frac{1}{2}}}dt,$ $-\pi<\mathop{\mathrm{ph}\/}\nolimits{t}<\pi$ , $c>0$ .

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{V\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\mathop{\mathrm{ph}\/}\nolimits$ : 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

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

§12.5(iii) Mellin–Barnes Integrals

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.
Keywords:: parabolic cylinder functions
Permalink:: http://dlmf.nist.gov/12.5.iii

12.5.8 $\mathop{U\/}\nolimits\!\left(a,z\right)=\frac{e^{{-\frac{1}{4}z^{2}}}z^{{-a-\frac{1}{2}}}}{2\pi i\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}+a\right)}\*\int _{{-i\infty}}^{{i\infty}}\mathop{\Gamma\/}\nolimits\!\left(t\right)\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}+a-2t\right)2^{t}z^{{2t}}dt,$ $a\neq-\frac{1}{2},-\frac{3}{2},-\frac{5}{2},\dots$ , $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{3}{4}\pi$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{U\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $z$ : complex variable and $a$ : real or complex parameter
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

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

12.5.9 $\mathop{V\/}\nolimits\!\left(a,z\right)=\sqrt{\frac{2}{\pi}}\frac{e^{{\frac{1}{4}z^{2}}}z^{{a-\frac{1}{2}}}}{2\pi i\mathop{\Gamma\/}\nolimits\!\left(\frac{1}{2}-a\right)}\*\int _{{-i\infty}}^{{i\infty}}\mathop{\Gamma\/}\nolimits\!\left(t\right)\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{2}-a-2t\right)2^{t}z^{{2t}}\mathop{\cos\/}\nolimits(\pi t)dt,$ $a\neq\frac{1}{2},\frac{3}{2},\frac{5}{2},\dots$ , $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{4}\pi$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{V\/}\nolimits\!\left(a,z\right)$ : parabolic cylinder function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $z$ : complex variable and $a$ : real or complex parameter
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

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

§12.5(iv) Compendia

Keywords:: parabolic cylinder functions
Permalink:: http://dlmf.nist.gov/12.5.iv

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).

§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