# §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 of $x$, $\operatorname{erf}\NVar{z}$: error function, $\mathrm{e}$: base of exponential function, $\int$: integral and $\Re$: real part 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 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 $\int_{0}^{\infty}e^{-at}C\left(t\right)\mathrm{d}t=\frac{1}{a}\mathrm{f}\left(% \frac{a}{\pi}\right),$ $\Re a>0$,
 7.14.6 $\int_{0}^{\infty}e^{-at}S\left(t\right)\mathrm{d}t=\frac{1}{a}\mathrm{g}\left(% \frac{a}{\pi}\right),$ $\Re a>0$,
 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 exponential function, $\int$: integral and $\Re$: real part 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 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 exponential function, $\int$: integral and $\Re$: real part 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 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.