§7.7 Integral Representations

Integrals of the type $\int {\mathrm{e}}^{-z^2}R(z)dz$, 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		$$\mathrm{erfc}z=\frac{2}{\pi }{\mathrm{e}}^{-z^2}{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-z^2{t}^2}}{t^2+1}dt,$$
$|\mathrm{ph}z|\le \frac{1}{4}\pi$,
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: 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		$$w\left(z\right)=\frac{1}{\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-t^2}dt}{t-z}=\frac{2z}{\pi \mathrm{i}}{\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-t^2}dt}{t^2-z^2},$$
$\mathrm{\Im }z>0$.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{e}$: base of natural logarithm, $\mathrm{\Im }$: 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		$${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-a{t}^2+2\mathrm{i}zt}dt=\frac{1}{2}\sqrt{\frac{\pi }{a}}{\mathrm{e}}^{-z^2/a}+\frac{\mathrm{i}}{\sqrt{a}}F\left(\frac{z}{\sqrt{a}}\right),$$
$\mathrm{\Re }a>0$.
ⓘ Symbols: $F\left(z\right)$: Dawson’s integral, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: 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		$${\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-at}}{\sqrt{t+z^2}}dt=\sqrt{\frac{\pi }{a}}{\mathrm{e}}^{a{z}^2}\mathrm{erfc}\left(\sqrt{a}z\right),$$
$\mathrm{\Re }a>0$, $\mathrm{\Re }z>0$.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: 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		$${\int }_0^1\frac{{\mathrm{e}}^{-a{t}^2}}{t^2+1}dt=\frac{\pi }{4}{\mathrm{e}}^a\left(1-{(\mathrm{erf}\sqrt{a})}^2\right),$$
$\mathrm{\Re }a>0$.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{erf}z$: error function, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: 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		$${\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{-(a{t}^2+2bt+c)}dt=\frac{1}{2}\sqrt{\frac{\pi }{a}}{\mathrm{e}}^{(b^2-ac)/a}\mathrm{erfc}\left(\sqrt{a}x+\frac{b}{\sqrt{a}}\right),$$
$\mathrm{\Re }a>0$.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: 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		$${\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{-a^2{t}^2-(b^2/t^2)}dt=\frac{\sqrt{\pi }}{4a}\left({\mathrm{e}}^{2ab}\mathrm{erfc}\left(ax+(b/x)\right)+{\mathrm{e}}^{-2ab}\mathrm{erfc}\left(ax-(b/x)\right)\right),$$
$x>0$, .
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase and $x$: real variable A&S Ref: 7.4.33 (in different form) Referenced by: §7.7(i) 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		$${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-a^2{t}^2-(b^2/t^2)}dt=\frac{\sqrt{\pi }}{2a}{\mathrm{e}}^{-2ab},$$
, .
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{ph}$: 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		$${\int }_0^x\mathrm{erf}tdt=x\mathrm{erf}x+\frac{1}{\sqrt{\pi }}\left({\mathrm{e}}^{-x^2}-1\right).$$
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{erf}z$: 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

7.7.10		$\mathrm{f}\left(z\right)$	$={\displaystyle \frac{1}{\pi \sqrt{2}}}{\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{{\mathrm{e}}^{-\pi z^{2}t/2}}{\sqrt{t}(t^2+1)}}dt,$
$|\mathrm{ph}z|\le \frac{1}{4}\pi$,
ⓘ Symbols: $\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: 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		$\mathrm{g}\left(z\right)$	$={\displaystyle \frac{1}{\pi \sqrt{2}}}{\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{\sqrt{t}{\mathrm{e}}^{-\pi z^{2}t/2}}{t^2+1}}dt,$
$|\mathrm{ph}z|\le \frac{1}{4}\pi$,
ⓘ Symbols: $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: 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		$$\mathrm{g}\left(z\right)+\mathrm{i}\mathrm{f}\left(z\right)={\mathrm{e}}^{-\pi \mathrm{i}{z}^2/2}{\int }_z^{\mathrm{\infty }}{\mathrm{e}}^{\pi \mathrm{i}{t}^2/2}dt.$$
ⓘ Symbols: $\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: 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

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).