7 Error Functions, Dawson’s and Fresnel Integrals Properties 7.13 Zeros 7.15 Sums

§7.14 Integrals

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Permalink:: http://dlmf.nist.gov/7.14
See also:: Annotations for Ch.7

§7.14(i) Error Functions

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Notes:: For (7.14.1) and (7.14.2) integrate by parts and apply (7.7.3), (7.7.6). (7.14.3) follows from (7.14.4) with $c=0$ . For (7.14.4) integrate by parts and apply (10.32.10), (10.39.2).
Permalink:: http://dlmf.nist.gov/7.14.i
See also:: Annotations for §7.14 and Ch.7

Fourier Transform

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Keywords:: Fourier transform, error functions, integrals
See also:: Annotations for §7.14(i), §7.14 and Ch.7

7.14.1		$\int_{0}^{\infty}e^{2iat}\operatorname{erfc}\left(bt\right)\,\mathrm{d}t={% \frac{1}{a\sqrt{\pi}}F\left(\frac{a}{b}\right)+\frac{i}{2a}\left(1-e^{-(a/b)^{% 2}}\right)},$
$a\in\mathbb{C}$ , $\|\operatorname{ph}b\|<\tfrac{1}{4}\pi$ .
ⓘ Symbols: $F\left(\NVar{z}\right)$ : Dawson’s integral, $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$ : complementary error function, $\mathbb{C}$ : complex plane, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral and $\operatorname{ph}$ : phase A&S Ref: 7.4.18 (in different form) Referenced by: §7.14(i) Permalink: http://dlmf.nist.gov/7.14.E1 Encodings: TeX, pMML, png See also: Annotations for §7.14(i), §7.14(i), §7.14 and Ch.7

When $a=0$ the limit is taken.

Laplace Transforms

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Keywords:: Laplace transforms, error functions, integrals
See also:: Annotations for §7.14(i), §7.14 and Ch.7

7.14.2		$\int_{0}^{\infty}e^{-at}\operatorname{erf}\left(bt\right)\,\mathrm{d}t=\frac{1% }{a}e^{a^{2}/(4b^{2})}\operatorname{erfc}\left(\frac{a}{2b}\right),$
$\Re a>0$ , $\|\operatorname{ph}b\|<\tfrac{1}{4}\pi$ ,
ⓘ 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 $x$ , $\operatorname{erf}\NVar{z}$ : error function, $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{ph}$ : phase and $\Re$ : real part Keywords: Laplace transform A&S Ref: 7.4.17 Referenced by: §7.14(i) Permalink: http://dlmf.nist.gov/7.14.E2 Encodings: TeX, pMML, png See also: Annotations for §7.14(i), §7.14(i), §7.14 and Ch.7

7.14.3		$\int_{0}^{\infty}e^{-at}\operatorname{erf}\sqrt{bt}\,\mathrm{d}t=\frac{1}{a}% \sqrt{\frac{b}{a+b}},$
$\Re a>0$ , $\Re b>0$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\operatorname{erf}\NVar{z}$ : error function, $\mathrm{e}$ : base of natural logarithm, $\int$ : integral and $\Re$ : real part Keywords: Laplace transform A&S Ref: 7.4.19 Referenced by: §7.14(i) Permalink: http://dlmf.nist.gov/7.14.E3 Encodings: TeX, pMML, png See also: Annotations for §7.14(i), §7.14(i), §7.14 and Ch.7

7.14.4		$\int_{0}^{\infty}e^{(a-b)t}\operatorname{erfc}\left(\sqrt{at}+\sqrt{\frac{c}{t% }}\right)\,\mathrm{d}t=\frac{e^{-2(\sqrt{ac}+\sqrt{bc})}}{\sqrt{b}(\sqrt{a}+% \sqrt{b})},$
$\|\operatorname{ph}a\|<\frac{1}{2}\pi$ , $\Re b>0$ , $\Re c\geq 0$ .
ⓘ 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, $\operatorname{ph}$ : phase and $\Re$ : real part Keywords: Laplace transform A&S Ref: 7.4.21 Referenced by: §3.5(ix), §7.14(i) Permalink: http://dlmf.nist.gov/7.14.E4 Encodings: TeX, pMML, png See also: Annotations for §7.14(i), §7.14(i), §7.14 and Ch.7

§7.14(ii) Fresnel Integrals

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Notes:: For (7.14.5) and (7.14.6) integrate by parts, and use (7.7.15) and (7.7.16). For (7.14.7) and (7.14.8) consider the integrals $\int_{0}^{\infty}e^{-at}\left(C\left(\sqrt{2t/\pi}\right)\pm iS\left(\sqrt{2t/% \pi}\right)\right)\,\mathrm{d}t$ and integrate by parts. The results are $1\Bigm{/}\left(a\sqrt{2(a\mp i)}\right)$ , from which (7.14.7) and (7.14.8) follow.
Permalink:: http://dlmf.nist.gov/7.14.ii
See also:: Annotations for §7.14 and Ch.7

Laplace Transforms

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Keywords:: Fresnel integrals, Laplace transforms, integrals
See also:: Annotations for §7.14(ii), §7.14 and Ch.7

7.14.5		$\int_{0}^{\infty}e^{-at}C\left(t\right)\,\mathrm{d}t=\frac{1}{a}\mathrm{f}% \left(\frac{a}{\pi}\right),$
$\Re a>0$ ,
ⓘ Symbols: $\mathrm{f}\left(\NVar{z}\right)$ : auxiliary function for Fresnel integrals, $C\left(\NVar{z}\right)$ : Fresnel 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, $\int$ : integral and $\Re$ : real part Keywords: Laplace transform A&S Ref: 7.4.27 (in different notation) Referenced by: §7.14(ii) Permalink: http://dlmf.nist.gov/7.14.E5 Encodings: TeX, pMML, png See also: Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7

7.14.6		$\int_{0}^{\infty}e^{-at}S\left(t\right)\,\mathrm{d}t=\frac{1}{a}\mathrm{g}% \left(\frac{a}{\pi}\right),$
$\Re a>0$ ,
ⓘ Symbols: $\mathrm{g}\left(\NVar{z}\right)$ : auxiliary function for Fresnel integrals, $S\left(\NVar{z}\right)$ : Fresnel 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, $\int$ : integral and $\Re$ : real part Keywords: Laplace transform A&S Ref: 7.4.28 (in different notation) Referenced by: §7.14(ii) Permalink: http://dlmf.nist.gov/7.14.E6 Encodings: TeX, pMML, png See also: Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7

7.14.7		$\int_{0}^{\infty}e^{-at}C\left(\sqrt{\frac{2t}{\pi}}\right)\,\mathrm{d}t=\frac% {(\sqrt{a^{2}+1}+a)^{\frac{1}{2}}}{2a\sqrt{a^{2}+1}},$
$\Re a>0$ ,
ⓘ Symbols: $C\left(\NVar{z}\right)$ : Fresnel 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, $\int$ : integral and $\Re$ : real part Keywords: Laplace transform A&S Ref: 7.4.29 in modified form Referenced by: §7.14(ii) Permalink: http://dlmf.nist.gov/7.14.E7 Encodings: TeX, pMML, png See also: Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7

7.14.8		$\int_{0}^{\infty}e^{-at}S\left(\sqrt{\frac{2t}{\pi}}\right)\,\mathrm{d}t=\frac% {(\sqrt{a^{2}+1}-a)^{\frac{1}{2}}}{2a\sqrt{a^{2}+1}},$
$\Re a>0$ .
ⓘ Symbols: $S\left(\NVar{z}\right)$ : Fresnel 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, $\int$ : integral and $\Re$ : real part Keywords: Laplace transform A&S Ref: 7.4.30 in modified form Referenced by: §7.14(ii) Permalink: http://dlmf.nist.gov/7.14.E8 Encodings: TeX, pMML, png See also: Annotations for §7.14(ii), §7.14(ii), §7.14 and Ch.7

§7.14(iii) Compendia

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Permalink:: http://dlmf.nist.gov/7.14.iii
See also:: Annotations for §7.14 and Ch.7

For collections of integrals see Apelblat (1983, pp. 131–146), Erdélyi et al. (1954a, vol. 1, pp. 40, 96, 176–177), Geller and Ng (1971), Gradshteyn and Ryzhik (2015, §§5.4 and 6.28–6.32), Marichev (1983, pp. 184–189), Ng and Geller (1969), Oberhettinger (1974, pp. 138–139, 142–143), Oberhettinger (1990, pp. 48–52, 155–158), Oberhettinger and Badii (1973, pp. 171–172, 179–181), Prudnikov et al. (1986b, vol. 2, pp. 30–36, 93–143), Prudnikov et al. (1992a, §§3.7–3.8), and Prudnikov et al. (1992b, §§3.7–3.8). In a series of ten papers Hadži (1968, 1969, 1970, 1972, 1973, 1975a, 1975b, 1976a, 1976b, 1978) gives many integrals containing error functions and Fresnel integrals, also in combination with the hypergeometric function, confluent hypergeometric functions, and generalized hypergeometric functions.

7.13 Zeros 7.15 Sums

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§7.14 Integrals

Contents

§7.14(i) Error Functions

Fourier Transform

Laplace Transforms

§7.14(ii) Fresnel Integrals

Laplace Transforms

§7.14(iii) Compendia

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