§7.7 Integral Representations

§7.7(i) Error Functions and Dawson’s Integral

Integrals of the type $\int e^{-z^{2}}R(z)\mathrm{d}z$, 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 $\operatorname{erfc}z=\frac{2}{\pi}e^{-z^{2}}\int_{0}^{\infty}\frac{e^{-z^{2}t^% {2}}}{t^{2}+1}\mathrm{d}t,$ $|\operatorname{ph}z|\leq\frac{1}{4}\pi$, ⓘ 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, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\operatorname{ph}$: 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 $w\left(z\right)=\frac{1}{\pi i}\int_{-\infty}^{\infty}\frac{e^{-t^{2}}\mathrm{% d}t}{t-z}=\frac{2z}{\pi i}\int_{0}^{\infty}\frac{e^{-t^{2}}\mathrm{d}t}{t^{2}-% z^{2}},$ $\Im z>0$.
 7.7.3 $\int_{0}^{\infty}e^{-at^{2}+2izt}\mathrm{d}t=\frac{1}{2}\sqrt{\frac{\pi}{a}}e^% {-z^{2}/a}+\frac{i}{\sqrt{a}}F\left(\frac{z}{\sqrt{a}}\right),$ $\Re a>0$. ⓘ Symbols: $F\left(\NVar{z}\right)$: Dawson’s integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\Re$: 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 $\int_{0}^{\infty}\frac{e^{-at}}{\sqrt{t+z^{2}}}\mathrm{d}t=\sqrt{\frac{\pi}{a}% }e^{az^{2}}\operatorname{erfc}\left(\sqrt{a}z\right),$ $\Re a>0$, $\Re z>0$.
 7.7.5 $\int_{0}^{1}\frac{e^{-at^{2}}}{t^{2}+1}\mathrm{d}t=\frac{\pi}{4}e^{a}\left(1-(% \operatorname{erf}\sqrt{a})^{2}\right),$ $\Re a>0$. ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential, $\operatorname{erf}\NVar{z}$: error function, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\Re$: 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 $\int_{x}^{\infty}e^{-(at^{2}+2bt+c)}\mathrm{d}t=\frac{1}{2}\sqrt{\frac{\pi}{a}% }e^{(b^{2}-ac)/a}\operatorname{erfc}\left(\sqrt{a}x+\frac{b}{\sqrt{a}}\right),$ $\Re a>0$. ⓘ 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, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\Re$: 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 $\int_{x}^{\infty}e^{-a^{2}t^{2}-(b^{2}/t^{2})}\mathrm{d}t=\frac{\sqrt{\pi}}{4a% }\left(e^{2ab}\operatorname{erfc}\left(ax+(b/x)\right)+e^{-2ab}\operatorname{% erfc}\left(ax-(b/x)\right)\right),$ $x>0$, $|\operatorname{ph}a|<\tfrac{1}{4}\pi$. ⓘ 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, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\operatorname{ph}$: 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 $\int_{0}^{\infty}e^{-a^{2}t^{2}-(b^{2}/t^{2})}\mathrm{d}t=\frac{\sqrt{\pi}}{2a% }e^{-2ab},$ $|\operatorname{ph}a|<\tfrac{1}{4}\pi$, $|\operatorname{ph}b|<\tfrac{1}{4}\pi$. ⓘ Symbols: $\pi$: 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 $\operatorname{ph}$: 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 $\int_{0}^{x}\operatorname{erf}t\mathrm{d}t=x\operatorname{erf}x+\frac{1}{\sqrt% {\pi}}\left(e^{-x^{2}}-1\right).$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential, $\operatorname{erf}\NVar{z}$: 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(ii) Auxiliary Functions

 7.7.10 $\displaystyle\mathrm{f}\left(z\right)$ $\displaystyle=\frac{1}{\pi\sqrt{2}}\int_{0}^{\infty}\frac{e^{-\pi z^{2}t/2}}{% \sqrt{t}(t^{2}+1)}\mathrm{d}t,$ $|\operatorname{ph}z|\leq\frac{1}{4}\pi$, ⓘ 7.7.11 $\displaystyle\mathrm{g}\left(z\right)$ $\displaystyle=\frac{1}{\pi\sqrt{2}}\int_{0}^{\infty}\frac{\sqrt{t}e^{-\pi z^{2% }t/2}}{t^{2}+1}\mathrm{d}t,$ $|\operatorname{ph}z|\leq\frac{1}{4}\pi$, ⓘ
 7.7.12 $\mathrm{g}\left(z\right)+i\mathrm{f}\left(z\right)=e^{-\pi iz^{2}/2}\int_{z}^{% \infty}e^{\pi it^{2}/2}\mathrm{d}t.$

Mellin–Barnes Integrals

 7.7.13 $\mathrm{f}\left(z\right)=\frac{(2\pi)^{-3/2}}{2\pi i}\int_{c-i\infty}^{c+i% \infty}\zeta^{-s}\Gamma\left(s\right)\Gamma\left(s+\tfrac{1}{2}\right)\*\Gamma% \left(s+\tfrac{3}{4}\right)\Gamma\left(\tfrac{1}{4}-s\right)\mathrm{d}s,$
 7.7.14 $\mathrm{g}\left(z\right)=\frac{(2\pi)^{-3/2}}{2\pi i}\int_{c-i\infty}^{c+i% \infty}\zeta^{-s}\Gamma\left(s\right)\Gamma\left(s+\tfrac{1}{2}\right)\*\Gamma% \left(s+\tfrac{1}{4}\right)\Gamma\left(\tfrac{3}{4}-s\right)\mathrm{d}s.$

In (7.7.13) and (7.7.14) the integration paths are straight lines, $\zeta=\frac{1}{16}\pi^{2}z^{4}$, and $c$ is a constant such that $0 in (7.7.13), and $0 in (7.7.14).

 7.7.15 $\int_{0}^{\infty}e^{-at}\cos\left(t^{2}\right)\mathrm{d}t=\sqrt{\frac{\pi}{2}}% \mathrm{f}\left(\frac{a}{\sqrt{2\pi}}\right),$ $\Re a>0$, ⓘ Symbols: $\mathrm{f}\left(\NVar{z}\right)$: auxiliary function for Fresnel integrals, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function, $\mathrm{d}\NVar{x}$: differential, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\Re$: real part Keywords: Laplace transform A&S Ref: 7.4.22 (in different notation) Referenced by: §7.14(ii) Permalink: http://dlmf.nist.gov/7.7.E15 Encodings: TeX, pMML, png See also: Annotations for §7.7(ii), §7.7(ii), §7.7 and Ch.7
 7.7.16 $\int_{0}^{\infty}e^{-at}\sin\left(t^{2}\right)\mathrm{d}t=\sqrt{\frac{\pi}{2}}% \mathrm{g}\left(\frac{a}{\sqrt{2\pi}}\right),$ $\Re a>0$. ⓘ Symbols: $\mathrm{g}\left(\NVar{z}\right)$: auxiliary function for Fresnel integrals, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\Re$: real part and $\sin\NVar{z}$: sine function Keywords: Laplace transform A&S Ref: 7.4.23 (in different notation) Referenced by: §7.14(ii) Permalink: http://dlmf.nist.gov/7.7.E16 Encodings: TeX, pMML, png See also: Annotations for §7.7(ii), §7.7(ii), §7.7 and Ch.7

§7.7(iii) Compendia

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).