§7.14 Integrals

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§7.14(i) Error Functions
§7.14(ii) Fresnel Integrals
§7.14(iii) Compendia

§7.14(i) Error Functions

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

¶ Fourier Transform

Keywords:: Fourier transform, error functions, integrals

7.14.1 $\int _{0}^{\infty}e^{{2iat}}\mathop{\mathrm{erfc}\/}\nolimits\!\left(bt\right)dt={\frac{1}{a\sqrt{\pi}}\mathop{F\/}\nolimits\!\left(\frac{a}{b}\right)+\frac{i}{2a}\left(1-e^{{-(a/b)^{2}}}\right)},$ $a\in\Complex$ , $|\mathop{\mathrm{ph}\/}\nolimits b|<\tfrac{1}{4}\pi$ .

Symbols:: $\mathop{F\/}\nolimits\!\left(z\right)$ : Dawson’s integral, $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function, $\Complex$ : complex plane (excluding infinity), $dx$ : differential of $x$ , $\in$ : element of, $e$ : base of exponential function, $\int$ : integral and $\mathop{\mathrm{ph}\/}\nolimits$ : phase
A&S Ref:: 7.4.18 (in different form)
Referenced by:: §7.14(i)
Permalink:: http://dlmf.nist.gov/7.14.E1
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When $a=0$ the limit is taken.

¶ Laplace Transforms

Keywords:: error functions, integrals

7.14.2 $\int _{0}^{\infty}e^{{-at}}\mathop{\mathrm{erf}\/}\nolimits\!\left(bt\right)dt=\frac{1}{a}e^{{a^{2}/(4b^{2})}}\mathop{\mathrm{erfc}\/}\nolimits\!\left(\frac{a}{2b}\right),$ $\realpart{a}>0$ , $|\mathop{\mathrm{ph}\/}\nolimits b|<\tfrac{1}{4}\pi$ ,

Symbols:: $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function, $dx$ : differential of $x$ , $\mathop{\mathrm{erf}\/}\nolimits z$ : error function, $e$ : base of exponential function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase and $\realpart{}$ : real part
A&S Ref:: 7.4.17
Referenced by:: §7.14(i)
Permalink:: http://dlmf.nist.gov/7.14.E2
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7.14.3 $\int _{0}^{\infty}e^{{-at}}\mathop{\mathrm{erf}\/}\nolimits\sqrt{bt}dt=\frac{1}{a}\sqrt{\frac{b}{a+b}},$ $\realpart{a}>0$ , $\realpart{b}>0$ ,

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:: 7.4.19
Referenced by:: §7.14(i)
Permalink:: http://dlmf.nist.gov/7.14.E3
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7.14.4 $\int _{0}^{\infty}e^{{(a-b)t}}\mathop{\mathrm{erfc}\/}\nolimits\!\left(\sqrt{at}+\sqrt{\frac{c}{t}}\right)dt=\frac{e^{{-2(\sqrt{ac}+\sqrt{bc})}}}{\sqrt{b}(\sqrt{a}+\sqrt{b})},$ $|\mathop{\mathrm{ph}\/}\nolimits a|<\frac{1}{2}\pi$ , $\realpart{b}>0$ , $\realpart{c}\geq 0$ .

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 $\realpart{}$ : real part
A&S Ref:: 7.4.21
Referenced by:: ¶ ‣ §3.5(ix), §7.14(i)
Permalink:: http://dlmf.nist.gov/7.14.E4
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§7.14(ii) Fresnel Integrals

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 $\int _{0}^{\infty}e^{{-at}}\left(\mathop{C\/}\nolimits\!\left(\sqrt{2t/\pi}\right)\pm i\mathop{S\/}\nolimits\!\left(\sqrt{2t/\pi}\right)\right)dt$ 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

¶ Laplace Transforms

Keywords:: Fresnel integrals, integrals

7.14.5 $\int _{0}^{\infty}e^{{-at}}\mathop{C\/}\nolimits\!\left(t\right)dt=\frac{1}{a}\mathop{\mathrm{f}\/}\nolimits\!\left(\frac{a}{\pi}\right),$ $\realpart{a}>0$ ,

Symbols:: $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, $\mathop{C\/}\nolimits\!\left(z\right)$ : Fresnel integral, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral and $\realpart{}$ : real part
A&S Ref:: 7.4.27 (in different notation)
Referenced by:: §7.14(ii)
Permalink:: http://dlmf.nist.gov/7.14.E5
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7.14.6 $\int _{0}^{\infty}e^{{-at}}\mathop{S\/}\nolimits\!\left(t\right)dt=\frac{1}{a}\mathop{\mathrm{g}\/}\nolimits\!\left(\frac{a}{\pi}\right),$ $\realpart{a}>0$ ,

Symbols:: $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, $\mathop{S\/}\nolimits\!\left(z\right)$ : Fresnel integral, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral and $\realpart{}$ : real part
A&S Ref:: 7.4.28 (in different notation)
Referenced by:: §7.14(ii)
Permalink:: http://dlmf.nist.gov/7.14.E6
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7.14.7 $\int _{0}^{\infty}e^{{-at}}\mathop{C\/}\nolimits\!\left(\sqrt{\frac{2t}{\pi}}\right)dt=\frac{(\sqrt{a^{2}+1}+a)^{{\frac{1}{2}}}}{2a\sqrt{a^{2}+1}},$ $\realpart{a}>0$ ,

Symbols:: $\mathop{C\/}\nolimits\!\left(z\right)$ : Fresnel integral, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral and $\realpart{}$ : real part
A&S Ref:: 7.4.29 in modified form
Referenced by:: §7.14(ii)
Permalink:: http://dlmf.nist.gov/7.14.E7
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7.14.8 $\int _{0}^{\infty}e^{{-at}}\mathop{S\/}\nolimits\!\left(\sqrt{\frac{2t}{\pi}}\right)dt=\frac{(\sqrt{a^{2}+1}-a)^{{\frac{1}{2}}}}{2a\sqrt{a^{2}+1}},$ $\realpart{a}>0$ .

Symbols:: $\mathop{S\/}\nolimits\!\left(z\right)$ : Fresnel integral, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral and $\realpart{}$ : real part
A&S Ref:: 7.4.30 in modified form
Referenced by:: §7.14(ii)
Permalink:: http://dlmf.nist.gov/7.14.E8
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§7.14(iii) Compendia

Permalink:: http://dlmf.nist.gov/7.14.iii

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 (2000, §§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.

§7.14 Integrals

Contents

§7.14(i) Error Functions

¶ Fourier Transform

¶ Laplace Transforms

§7.14(ii) Fresnel Integrals

¶ Laplace Transforms

§7.14(iii) Compendia