§7.14 Integrals

§7.14(i) Error Functions

Fourier Transform

 7.14.1 $\int_{0}^{\infty}e^{2iat}\operatorname{erfc}\left(bt\right)\,\mathrm{d}t={% \frac{1}{a\sqrt{\pi}}F\left(\frac{a}{b}\right)+\frac{i}{2a}\left(1-e^{-(a/b)^{% 2}}\right)},$ $a\in\mathbb{C}$, $|\operatorname{ph}b|<\tfrac{1}{4}\pi$.

When $a=0$ the limit is taken.

Laplace Transforms

 7.14.2 $\int_{0}^{\infty}e^{-at}\operatorname{erf}\left(bt\right)\,\mathrm{d}t=\frac{1% }{a}e^{a^{2}/(4b^{2})}\operatorname{erfc}\left(\frac{a}{2b}\right),$ $\Re a>0$, $|\operatorname{ph}b|<\tfrac{1}{4}\pi$,
 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, $\operatorname{erf}\NVar{z}$: 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 $\int_{0}^{\infty}e^{(a-b)t}\operatorname{erfc}\left(\sqrt{at}+\sqrt{\frac{c}{t% }}\right)\,\mathrm{d}t=\frac{e^{-2(\sqrt{ac}+\sqrt{bc})}}{\sqrt{b}(\sqrt{a}+% \sqrt{b})},$ $|\operatorname{ph}a|<\frac{1}{2}\pi$, $\Re b>0$, $\Re c\geq 0$.

§7.14(ii) Fresnel Integrals

Laplace Transforms

 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, $\mathrm{e}$: 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, $\mathrm{e}$: 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, $\mathrm{e}$: 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, $\mathrm{e}$: 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

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.