# §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 $\operatorname{erfc}z=\frac{2}{\pi}e^{-z^{2}}\int_{0}^{\infty}\frac{e^{-z^{2}t^% {2}}}{t^{2}+1}\,\mathrm{d}t,$ $|\operatorname{ph}z|\leq\frac{1}{4}\pi$,
 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 $\int_{0}^{\infty}e^{-at^{2}+2izt}\,\mathrm{d}t=\frac{1}{2}\sqrt{\frac{\pi}{a}}% e^{-z^{2}/a}+\frac{i}{\sqrt{a}}F\left(\frac{z}{\sqrt{a}}\right),$ $\Re a>0$.
 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 $\int_{0}^{1}\frac{e^{-at^{2}}}{t^{2}+1}\,\mathrm{d}t=\frac{\pi}{4}e^{a}\left(1% -(\operatorname{erf}\sqrt{a})^{2}\right),$ $\Re a>0$.
 7.7.6 $\int_{x}^{\infty}e^{-(at^{2}+2bt+c)}\,\mathrm{d}t=\frac{1}{2}\sqrt{\frac{\pi}{% a}}e^{(b^{2}-ac)/a}\operatorname{erfc}\left(\sqrt{a}x+\frac{b}{\sqrt{a}}\right),$ $\Re a>0$.
 7.7.7 $\int_{x}^{\infty}e^{-a^{2}t^{2}-(b^{2}/t^{2})}\,\mathrm{d}t=\frac{\sqrt{\pi}}{% 4a}\left(e^{2ab}\operatorname{erfc}\left(ax+(b/x)\right)+e^{-2ab}\operatorname% {erfc}\left(ax-(b/x)\right)\right),$ $x>0$, $|\operatorname{ph}a|<\tfrac{1}{4}\pi$.
 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 of $x$, $\mathrm{e}$: 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 of $x$, $\operatorname{erf}\NVar{z}$: 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 $\int_{0}^{\infty}e^{-at}\cos\left(t^{2}\right)\,\mathrm{d}t=\sqrt{\frac{\pi}{2% }}\mathrm{f}\left(\frac{a}{\sqrt{2\pi}}\right),$ $\Re a>0$,
 7.7.16 $\int_{0}^{\infty}e^{-at}\sin\left(t^{2}\right)\,\mathrm{d}t=\sqrt{\frac{\pi}{2% }}\mathrm{g}\left(\frac{a}{\sqrt{2\pi}}\right),$ $\Re a>0$.

## §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).