7 Error Functions, Dawson’s and Fresnel Integrals Properties 7.6 Series Expansions 7.8 Inequalities

§7.7 Integral Representations

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Permalink:: http://dlmf.nist.gov/7.7
See also:: Annotations for Ch.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

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Keywords:: Dawson’s integral, error functions, integral representation, integral representations
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).
Permalink:: http://dlmf.nist.gov/7.7.i
See also:: Annotations for §7.7 and Ch.7

Integrals of the type $\int e^{-z^{2}}R(z)\,\mathrm{d}z$ , 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		$\operatorname{erfc}z=\frac{2}{\pi}e^{-z^{2}}\int_{0}^{\infty}\frac{e^{-z^{2}t^% {2}}}{t^{2}+1}\,\mathrm{d}t,$
$\|\operatorname{ph}z\|\leq\frac{1}{4}\pi$ ,
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{ph}$ : 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 Encodings: TeX, pMML, png See also: Annotations for §7.7(i), §7.7 and Ch.7

7.7.2		$w\left(z\right)=\frac{1}{\pi i}\int_{-\infty}^{\infty}\frac{e^{-t^{2}}\,% \mathrm{d}t}{t-z}=\frac{2z}{\pi i}\int_{0}^{\infty}\frac{e^{-t^{2}}\,\mathrm{d% }t}{t^{2}-z^{2}},$
$\Im z>0$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $w\left(\NVar{z}\right)$ : Faddeeva (or Faddeyeva) function, $\mathrm{e}$ : base of natural logarithm, $\Im$ : imaginary part, $\mathrm{i}$ : imaginary unit, $\int$ : 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 Encodings: TeX, pMML, png See also: Annotations for §7.7(i), §7.7 and Ch.7

7.7.3		$\int_{0}^{\infty}e^{-at^{2}+2izt}\,\mathrm{d}t=\frac{1}{2}\sqrt{\frac{\pi}{a}}% e^{-z^{2}/a}+\frac{i}{\sqrt{a}}F\left(\frac{z}{\sqrt{a}}\right),$
$\Re a>0$ .
ⓘ Symbols: $F\left(\NVar{z}\right)$ : Dawson’s integral, $\pi$ : 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, $\Re$ : 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 Encodings: TeX, pMML, png See also: Annotations for §7.7(i), §7.7 and Ch.7

7.7.4		$\int_{0}^{\infty}\frac{e^{-at}}{\sqrt{t+z^{2}}}\,\mathrm{d}t=\sqrt{\frac{\pi}{% a}}e^{az^{2}}\operatorname{erfc}\left(\sqrt{a}z\right),$
$\Re a>0$ , $\Re z>0$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part and $z$ : complex variable Keywords: Laplace transform A&S Ref: 7.4.8 Referenced by: §7.7(i) Permalink: http://dlmf.nist.gov/7.7.E4 Encodings: TeX, pMML, png See also: Annotations for §7.7(i), §7.7 and Ch.7

7.7.5		$\int_{0}^{1}\frac{e^{-at^{2}}}{t^{2}+1}\,\mathrm{d}t=\frac{\pi}{4}e^{a}\left(1% -(\operatorname{erf}\sqrt{a})^{2}\right),$
$\Re a>0$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{erf}\NVar{z}$ : error function, $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and $\Re$ : real part A&S Ref: 7.4.12 Referenced by: §7.7(i) Permalink: http://dlmf.nist.gov/7.7.E5 Encodings: TeX, pMML, png See also: Annotations for §7.7(i), §7.7 and Ch.7

7.7.6		$\int_{x}^{\infty}e^{-(at^{2}+2bt+c)}\,\mathrm{d}t=\frac{1}{2}\sqrt{\frac{\pi}{% a}}e^{(b^{2}-ac)/a}\operatorname{erfc}\left(\sqrt{a}x+\frac{b}{\sqrt{a}}\right),$
$\Re a>0$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : 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 Encodings: TeX, pMML, png See also: Annotations for §7.7(i), §7.7 and Ch.7

7.7.7		$\int_{x}^{\infty}e^{-a^{2}t^{2}-(b^{2}/t^{2})}\,\mathrm{d}t=\frac{\sqrt{\pi}}{% 4a}\left(e^{2ab}\operatorname{erfc}\left(ax+(b/x)\right)+e^{-2ab}\operatorname% {erfc}\left(ax-(b/x)\right)\right),$
$x>0$ , $\|\operatorname{ph}a\|<\tfrac{1}{4}\pi$ .
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{ph}$ : phase and $x$ : real variable A&S Ref: 7.4.33 (in different form) Referenced by: §7.7(i), §7.8 Permalink: http://dlmf.nist.gov/7.7.E7 Encodings: TeX, pMML, png See also: Annotations for §7.7(i), §7.7 and Ch.7

7.7.8		$\int_{0}^{\infty}e^{-a^{2}t^{2}-(b^{2}/t^{2})}\,\mathrm{d}t=\frac{\sqrt{\pi}}{% 2a}e^{-2ab},$
$\|\operatorname{ph}a\|<\tfrac{1}{4}\pi$ , $\|\operatorname{ph}b\|<\tfrac{1}{4}\pi$ .
ⓘ 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, $\int$ : integral and $\operatorname{ph}$ : phase A&S Ref: 7.4.3 (in different form) Referenced by: §7.7(i) Permalink: http://dlmf.nist.gov/7.7.E8 Encodings: TeX, pMML, png See also: Annotations for §7.7(i), §7.7 and Ch.7

7.7.9		$\int_{0}^{x}\operatorname{erf}t\,\mathrm{d}t=x\operatorname{erf}x+\frac{1}{% \sqrt{\pi}}\left(e^{-x^{2}}-1\right).$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{erf}\NVar{z}$ : error function, $\mathrm{e}$ : base of natural logarithm, $\int$ : 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 Encodings: TeX, pMML, png See also: Annotations for §7.7(i), §7.7 and Ch.7

§7.7(ii) Auxiliary Functions

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Keywords:: auxiliary functions for Fresnel integrals, integral representations
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).
Permalink:: http://dlmf.nist.gov/7.7.ii
See also:: Annotations for §7.7 and Ch.7

7.7.10	$\displaystyle\mathrm{f}\left(z\right)$	$\displaystyle=\frac{1}{\pi\sqrt{2}}\int_{0}^{\infty}\frac{e^{-\pi z^{2}t/2}}{% \sqrt{t}(t^{2}+1)}\,\mathrm{d}t,$
$\|\operatorname{ph}z\|\leq\frac{1}{4}\pi$ ,
ⓘ Symbols: $\mathrm{f}\left(\NVar{z}\right)$ : auxiliary function for Fresnel integrals, $\pi$ : 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, $\operatorname{ph}$ : 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 Encodings: TeX, pMML, png See also: Annotations for §7.7(ii), §7.7 and Ch.7
7.7.11	$\displaystyle\mathrm{g}\left(z\right)$	$\displaystyle=\frac{1}{\pi\sqrt{2}}\int_{0}^{\infty}\frac{\sqrt{t}e^{-\pi z^{2% }t/2}}{t^{2}+1}\,\mathrm{d}t,$
$\|\operatorname{ph}z\|\leq\frac{1}{4}\pi$ ,
ⓘ Symbols: $\mathrm{g}\left(\NVar{z}\right)$ : auxiliary function for Fresnel integrals, $\pi$ : 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, $\operatorname{ph}$ : 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 Encodings: TeX, pMML, png See also: Annotations for §7.7(ii), §7.7 and Ch.7

7.7.12		$\mathrm{g}\left(z\right)+i\mathrm{f}\left(z\right)=e^{-\pi iz^{2}/2}\int_{z}^{% \infty}e^{\pi it^{2}/2}\,\mathrm{d}t.$
ⓘ Symbols: $\mathrm{f}\left(\NVar{z}\right)$ : auxiliary function for Fresnel integrals, $\mathrm{g}\left(\NVar{z}\right)$ : auxiliary function for Fresnel integrals, $\pi$ : 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 and $z$ : complex variable Referenced by: §7.7(ii) Permalink: http://dlmf.nist.gov/7.7.E12 Encodings: TeX, pMML, png See also: Annotations for §7.7(ii), §7.7 and Ch.7

Mellin–Barnes Integrals

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Keywords:: Mellin–Barnes integrals, auxiliary functions for Fresnel integrals, integral representations
See also:: Annotations for §7.7(ii), §7.7 and Ch.7

7.7.13		$\mathrm{f}\left(z\right)=\frac{(2\pi)^{-3/2}}{2\pi i}\int_{c-i\infty}^{c+i% \infty}\zeta^{-s}\Gamma\left(s\right)\Gamma\left(s+\tfrac{1}{2}\right)\*\Gamma% \left(s+\tfrac{3}{4}\right)\Gamma\left(\tfrac{1}{4}-s\right)\,\mathrm{d}s,$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathrm{f}\left(\NVar{z}\right)$ : auxiliary function for Fresnel integrals, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $z$ : complex variable, $\zeta$ : change of variable and $c$ : constant Keywords: Mellin transform Referenced by: §7.7(ii), §7.7(ii) Permalink: http://dlmf.nist.gov/7.7.E13 Encodings: TeX, pMML, png See also: Annotations for §7.7(ii), §7.7(ii), §7.7 and Ch.7

7.7.14		$\mathrm{g}\left(z\right)=\frac{(2\pi)^{-3/2}}{2\pi i}\int_{c-i\infty}^{c+i% \infty}\zeta^{-s}\Gamma\left(s\right)\Gamma\left(s+\tfrac{1}{2}\right)\*\Gamma% \left(s+\tfrac{1}{4}\right)\Gamma\left(\tfrac{3}{4}-s\right)\,\mathrm{d}s.$
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\mathrm{g}\left(\NVar{z}\right)$ : auxiliary function for Fresnel integrals, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $z$ : complex variable, $\zeta$ : change of variable and $c$ : constant Keywords: Mellin transform Referenced by: §7.7(ii), §7.7(ii) Permalink: http://dlmf.nist.gov/7.7.E14 Encodings: TeX, pMML, png See also: Annotations for §7.7(ii), §7.7(ii), §7.7 and Ch.7

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		$\int_{0}^{\infty}e^{-at}\cos\left(t^{2}\right)\,\mathrm{d}t=\sqrt{\frac{\pi}{2% }}\mathrm{f}\left(\frac{a}{\sqrt{2\pi}}\right),$
$\Re a>0$ ,
ⓘ Symbols: $\mathrm{f}\left(\NVar{z}\right)$ : auxiliary function for Fresnel integrals, $\pi$ : 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 and $\Re$ : real part Keywords: Laplace transform A&S Ref: 7.4.22 (in different notation) Referenced by: §7.14(ii) Permalink: http://dlmf.nist.gov/7.7.E15 Encodings: TeX, pMML, png See also: Annotations for §7.7(ii), §7.7(ii), §7.7 and Ch.7

7.7.16		$\int_{0}^{\infty}e^{-at}\sin\left(t^{2}\right)\,\mathrm{d}t=\sqrt{\frac{\pi}{2% }}\mathrm{g}\left(\frac{a}{\sqrt{2\pi}}\right),$
$\Re a>0$ .
ⓘ Symbols: $\mathrm{g}\left(\NVar{z}\right)$ : auxiliary function for Fresnel integrals, $\pi$ : 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, $\Re$ : real part and $\sin\NVar{z}$ : sine function Keywords: Laplace transform A&S Ref: 7.4.23 (in different notation) Referenced by: §7.14(ii) Permalink: http://dlmf.nist.gov/7.7.E16 Encodings: TeX, pMML, png See also: Annotations for §7.7(ii), §7.7(ii), §7.7 and Ch.7

§7.7(iii) Compendia

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Permalink:: http://dlmf.nist.gov/7.7.iii
See also:: Annotations for §7.7 and Ch.7

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).

7.6 Series Expansions 7.8 Inequalities

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§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