§7.7 Integral Representations

Permalink:: http://dlmf.nist.gov/7.7

§7.7(i) Error Functions and Dawson’s Integral
§7.7(ii) Auxiliary Functions
§7.7(iii) Compendia

§7.7(i) Error Functions and Dawson’s Integral

Notes:: (7.7.1), (7.7.2), and (7.7.4) are given in van der Laan and Temme (1984, pp. 185–186). (7.7.3) follows from integrating from 0 to $iz/a$ and from $iz/a$ to $\infty$ . (7.7.5) follows by differentiating with respect to $a$ (after multiplying the equation by $e^{{-a}}$ ). (7.7.6), (7.7.7), and (7.7.9) follow by differentiating with respect to $x$ . For (7.7.8) let $x\to 0+$ in (7.7.7) and use (7.2.2), (7.2.4).
Keywords:: Dawson’s integral, error functions
Permalink:: http://dlmf.nist.gov/7.7.i

Integrals of the type $\int e^{{-z^{2}}}R(z)dz$ , where $R(z)$ is an arbitrary rational function, can be written in closed form in terms of the error functions and elementary functions.

7.7.1 $\mathop{\mathrm{erfc}\/}\nolimits z=\frac{2}{\pi}e^{{-z^{2}}}\int _{0}^{\infty}\frac{e^{{-z^{2}t^{2}}}}{t^{2}+1}dt,$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\frac{1}{4}\pi$ ,

Symbols:: $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase and $z$ : complex variable
A&S Ref:: 7.4.11 (in different form)
Referenced by:: §7.11, §7.7(i)
Permalink:: http://dlmf.nist.gov/7.7.E1
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7.7.2 $\mathop{w\/}\nolimits\!\left(z\right)=\frac{1}{\pi i}\int _{{-\infty}}^{\infty}\frac{e^{{-t^{2}}}dt}{t-z}=\frac{2z}{\pi i}\int _{0}^{\infty}\frac{e^{{-t^{2}}}dt}{t^{2}-z^{2}},$ $\imagpart{z}>0$ .

Symbols:: $dx$ : differential of $x$ , $\mathop{w\/}\nolimits\!\left(z\right)$ : complementary error function, $e$ : base of exponential function, $\imagpart{}$ : imaginary part, $\int$ : integral and $z$ : complex variable
A&S Ref:: 7.1.4 (in different form)
Referenced by:: §7.19(i), §7.22(i), §7.7(i)
Permalink:: http://dlmf.nist.gov/7.7.E2
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7.7.3 $\int _{0}^{\infty}e^{{-at^{2}+2izt}}dt=\frac{1}{2}\sqrt{\frac{\pi}{a}}e^{{-z^{2}/a}}+\frac{i}{\sqrt{a}}\mathop{F\/}\nolimits\!\left(\frac{z}{\sqrt{a}}\right),$ $\realpart{a}>0$ .

Symbols:: $\mathop{F\/}\nolimits\!\left(z\right)$ : Dawson’s integral, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part and $z$ : complex variable
A&S Ref:: 7.4.6 7.4.7 (in different notation)
Referenced by:: §7.14(i), §7.7(i)
Permalink:: http://dlmf.nist.gov/7.7.E3
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7.7.4 $\int _{0}^{\infty}\frac{e^{{-at}}}{\sqrt{t+z^{2}}}dt=\sqrt{\frac{\pi}{a}}e^{{az^{2}}}\mathop{\mathrm{erfc}\/}\nolimits\!\left(\sqrt{a}z\right),$ $\realpart{a}>0$ , $\realpart{z}>0$ .

Symbols:: $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part and $z$ : complex variable
A&S Ref:: 7.4.8
Referenced by:: §7.7(i)
Permalink:: http://dlmf.nist.gov/7.7.E4
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7.7.5 $\int _{0}^{1}\frac{e^{{-at^{2}}}}{t^{2}+1}dt=\frac{\pi}{4}e^{a}\left(1-(\mathop{\mathrm{erf}\/}\nolimits\sqrt{a})^{2}\right),$ $\realpart{a}>0$ .

Symbols:: $dx$ : differential of $x$ , $\mathop{\mathrm{erf}\/}\nolimits z$ : error function, $e$ : base of exponential function, $\int$ : integral and $\realpart{}$ : real part
A&S Ref:: 7.4.12
Referenced by:: §7.7(i)
Permalink:: http://dlmf.nist.gov/7.7.E5
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7.7.6 $\int _{x}^{\infty}e^{{-(at^{2}+2bt+c)}}dt=\frac{1}{2}\sqrt{\frac{\pi}{a}}e^{{(b^{2}-ac)/a}}\mathop{\mathrm{erfc}\/}\nolimits\!\left(\sqrt{a}x+\frac{b}{\sqrt{a}}\right),$ $\realpart{a}>0$ .

Symbols:: $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part and $x$ : real variable
A&S Ref:: 7.4.32 (in different form)
Referenced by:: §7.14(i), §7.7(i)
Permalink:: http://dlmf.nist.gov/7.7.E6
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7.7.7 $\int _{x}^{\infty}e^{{-a^{2}t^{2}-(b^{2}/t^{2})}}dt=\frac{\sqrt{\pi}}{4a}\left(e^{{2ab}}\mathop{\mathrm{erfc}\/}\nolimits\!\left(ax+(b/x)\right)+e^{{-2ab}}\mathop{\mathrm{erfc}\/}\nolimits\!\left(ax-(b/x)\right)\right),$ $x>0$ , $|\mathop{\mathrm{ph}\/}\nolimits a|<\tfrac{1}{4}\pi$ .

Symbols:: $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase and $x$ : real variable
A&S Ref:: 7.4.33 (in different form)
Referenced by:: §7.7(i)
Permalink:: http://dlmf.nist.gov/7.7.E7
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7.7.8 $\int _{0}^{\infty}e^{{-a^{2}t^{2}-(b^{2}/t^{2})}}dt=\frac{\sqrt{\pi}}{2a}e^{{-2ab}},$ $|\mathop{\mathrm{ph}\/}\nolimits a|<\tfrac{1}{4}\pi$ , $|\mathop{\mathrm{ph}\/}\nolimits b|<\tfrac{1}{4}\pi$ .

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral and $\mathop{\mathrm{ph}\/}\nolimits$ : phase
A&S Ref:: 7.4.3 (in different form)
Referenced by:: §7.7(i)
Permalink:: http://dlmf.nist.gov/7.7.E8
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7.7.9 $\int _{0}^{x}\mathop{\mathrm{erf}\/}\nolimits tdt=x\mathop{\mathrm{erf}\/}\nolimits x+\frac{1}{\sqrt{\pi}}\left(e^{{-x^{2}}}-1\right).$

Symbols:: $dx$ : differential of $x$ , $\mathop{\mathrm{erf}\/}\nolimits z$ : error function, $e$ : base of exponential function, $\int$ : integral and $x$ : real variable
A&S Ref:: 7.4.35 (in different form)
Referenced by:: §7.7(i)
Permalink:: http://dlmf.nist.gov/7.7.E9
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§7.7(ii) Auxiliary Functions

Notes:: For (7.7.10) and (7.7.11) see van der Laan and Temme (1984, Chapter V). (7.7.12) follows from (7.5.5) and (7.2.6). (7.7.13) and (7.7.14) follow by taking Mellin transforms (§1.14(iv)), and applying (7.7.10), (7.7.11), (5.12.1).
Keywords:: auxiliary functions for Fresnel integrals
Permalink:: http://dlmf.nist.gov/7.7.ii

7.7.10 $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)=\frac{1}{\pi\sqrt{2}}\int _{0}^{\infty}\frac{e^{{-\pi z^{2}t/2}}}{\sqrt{t}(t^{2}+1)}dt,$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\frac{1}{4}\pi$ ,

Symbols:: $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase and $z$ : complex variable
A&S Ref:: 7.4.26 (in different form)
Referenced by:: §7.12(ii), §7.7(ii)
Permalink:: http://dlmf.nist.gov/7.7.E10
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7.7.11 $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)=\frac{1}{\pi\sqrt{2}}\int _{0}^{\infty}\frac{\sqrt{t}e^{{-\pi z^{2}t/2}}}{t^{2}+1}dt,$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\frac{1}{4}\pi$ ,

Symbols:: $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase and $z$ : complex variable
A&S Ref:: 7.4.25 (in different form)
Referenced by:: §7.12(ii), §7.7(ii)
Permalink:: http://dlmf.nist.gov/7.7.E11
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7.7.12 $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)+i\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)=e^{{-\pi iz^{2}/2}}\int _{z}^{\infty}e^{{\pi it^{2}/2}}dt.$

Symbols:: $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral and $z$ : complex variable
Referenced by:: §7.7(ii)
Permalink:: http://dlmf.nist.gov/7.7.E12
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¶ Mellin–Barnes Integrals

Keywords:: Mellin–Barnes integrals, auxiliary functions for Fresnel integrals

7.7.13 $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)=\frac{(2\pi)^{{-3/2}}}{2\pi i}\int _{{c-i\infty}}^{{c+i\infty}}\zeta^{{-s}}\mathop{\Gamma\/}\nolimits\!\left(s\right)\mathop{\Gamma\/}\nolimits\!\left(s+\tfrac{1}{2}\right)\*\mathop{\Gamma\/}\nolimits\!\left(s+\tfrac{3}{4}\right)\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{4}-s\right)ds,$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, $dx$ : differential of $x$ , $\int$ : integral, $z$ : complex variable, $\zeta$ : change of variable and $c$ : constant
Referenced by:: ¶ ‣ §7.7(ii), §7.7(ii)
Permalink:: http://dlmf.nist.gov/7.7.E13
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7.7.14 $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)=\frac{(2\pi)^{{-3/2}}}{2\pi i}\int _{{c-i\infty}}^{{c+i\infty}}\zeta^{{-s}}\mathop{\Gamma\/}\nolimits\!\left(s\right)\mathop{\Gamma\/}\nolimits\!\left(s+\tfrac{1}{2}\right)\*\mathop{\Gamma\/}\nolimits\!\left(s+\tfrac{1}{4}\right)\mathop{\Gamma\/}\nolimits\!\left(\tfrac{3}{4}-s\right)ds.$

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, $dx$ : differential of $x$ , $\int$ : integral, $z$ : complex variable, $\zeta$ : change of variable and $c$ : constant
Referenced by:: ¶ ‣ §7.7(ii), §7.7(ii)
Permalink:: http://dlmf.nist.gov/7.7.E14
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In (7.7.13) and (7.7.14) the integration paths are straight lines, $\zeta=\frac{1}{16}\pi^{2}z^{4}$ , and $c$ is a constant such that $0<c<\frac{1}{4}$ in (7.7.13), and $0<c<\frac{3}{4}$ in (7.7.14).

7.7.15 $\int _{0}^{\infty}e^{{-at}}\mathop{\cos\/}\nolimits\!\left(t^{2}\right)dt=\sqrt{\frac{\pi}{2}}\mathop{\mathrm{f}\/}\nolimits\!\left(\frac{a}{\sqrt{2\pi}}\right),$ $\realpart{a}>0$ ,

Symbols:: $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral and $\realpart{}$ : real part
A&S Ref:: 7.4.22 (in different notation)
Referenced by:: §7.14(ii)
Permalink:: http://dlmf.nist.gov/7.7.E15
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7.7.16 $\int _{0}^{\infty}e^{{-at}}\mathop{\sin\/}\nolimits\!\left(t^{2}\right)dt=\sqrt{\frac{\pi}{2}}\mathop{\mathrm{g}\/}\nolimits\!\left(\frac{a}{\sqrt{2\pi}}\right),$ $\realpart{a}>0$ .

Symbols:: $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part and $\mathop{\sin\/}\nolimits z$ : sine function
A&S Ref:: 7.4.23 (in different notation)
Referenced by:: §7.14(ii)
Permalink:: http://dlmf.nist.gov/7.7.E16
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§7.7(iii) Compendia

Permalink:: http://dlmf.nist.gov/7.7.iii

For other integral representations see Erdélyi et al. (1954a, vol. 1, pp. 265–267, 270), Ng and Geller (1969), Oberhettinger (1974, pp. 246–248), and Oberhettinger and Badii (1973, pp. 371–377).

§7.7 Integral Representations

Contents

§7.7(i) Error Functions and Dawson’s Integral

§7.7(ii) Auxiliary Functions

¶ Mellin–Barnes Integrals

§7.7(iii) Compendia