# §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 $\mathop{\mathrm{erfc}\/}\nolimits z=\frac{2}{\pi}e^{-z^{2}}\int_{0}^{\infty}% \frac{e^{-z^{2}t^{2}}}{t^{2}+1}\mathrm{d}t,$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\frac{1}{4}\pi$,
 7.7.2 $\mathop{w\/}\nolimits\!\left(z\right)=\frac{1}{\pi i}\int_{-\infty}^{\infty}% \frac{e^{-t^{2}}\mathrm{d}t}{t-z}=\frac{2z}{\pi i}\int_{0}^{\infty}\frac{e^{-t% ^{2}}\mathrm{d}t}{t^{2}-z^{2}},$ $\Im{z}>0$.
 7.7.3 $\int_{0}^{\infty}e^{-at^{2}+2izt}\mathrm{d}t=\frac{1}{2}\sqrt{\frac{\pi}{a}}e^% {-z^{2}/a}+\frac{i}{\sqrt{a}}\mathop{F\/}\nolimits\!\left(\frac{z}{\sqrt{a}}% \right),$ $\Re{a}>0$.
 7.7.4 $\int_{0}^{\infty}\frac{e^{-at}}{\sqrt{t+z^{2}}}\mathrm{d}t=\sqrt{\frac{\pi}{a}% }e^{az^{2}}\mathop{\mathrm{erfc}\/}\nolimits\!\left(\sqrt{a}z\right),$ $\Re{a}>0$, $\Re{z}>0$.
 7.7.5 $\int_{0}^{1}\frac{e^{-at^{2}}}{t^{2}+1}\mathrm{d}t=\frac{\pi}{4}e^{a}\left(1-(% \mathop{\mathrm{erf}\/}\nolimits\sqrt{a})^{2}\right),$ $\Re{a}>0$.
 7.7.6 $\int_{x}^{\infty}e^{-(at^{2}+2bt+c)}\mathrm{d}t=\frac{1}{2}\sqrt{\frac{\pi}{a}% }e^{(b^{2}-ac)/a}\mathop{\mathrm{erfc}\/}\nolimits\!\left(\sqrt{a}x+\frac{b}{% \sqrt{a}}\right),$ $\Re{a}>0$.
 7.7.7 $\int_{x}^{\infty}e^{-a^{2}t^{2}-(b^{2}/t^{2})}\mathrm{d}t=\frac{\sqrt{\pi}}{4a% }\left(e^{2ab}\mathop{\mathrm{erfc}\/}\nolimits\!\left(ax+(b/x)\right)+e^{-2ab% }\mathop{\mathrm{erfc}\/}\nolimits\!\left(ax-(b/x)\right)\right),$ $x>0$, $|\mathop{\mathrm{ph}\/}\nolimits a|<\tfrac{1}{4}\pi$.
 7.7.8 $\int_{0}^{\infty}e^{-a^{2}t^{2}-(b^{2}/t^{2})}\mathrm{d}t=\frac{\sqrt{\pi}}{2a% }e^{-2ab},$ $|\mathop{\mathrm{ph}\/}\nolimits a|<\tfrac{1}{4}\pi$, $|\mathop{\mathrm{ph}\/}\nolimits b|<\tfrac{1}{4}\pi$. Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential of $x$, $\mathrm{e}$: base of exponential function, $\int$: integral and $\mathop{\mathrm{ph}\/}\nolimits$: 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)
 7.7.9 $\int_{0}^{x}\mathop{\mathrm{erf}\/}\nolimits t\mathrm{d}t=x\mathop{\mathrm{erf% }\/}\nolimits x+\frac{1}{\sqrt{\pi}}\left(e^{-x^{2}}-1\right).$ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential of $x$, $\mathop{\mathrm{erf}\/}\nolimits\NVar{z}$: error function, $\mathrm{e}$: base of exponential function, $\int$: 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(ii) Auxiliary Functions

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

### Mellin–Barnes Integrals

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

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 $\int_{0}^{\infty}e^{-at}\mathop{\cos\/}\nolimits\!\left(t^{2}\right)\mathrm{d}% t=\sqrt{\frac{\pi}{2}}\mathop{\mathrm{f}\/}\nolimits\!\left(\frac{a}{\sqrt{2% \pi}}\right),$ $\Re{a}>0$,
 7.7.16 $\int_{0}^{\infty}e^{-at}\mathop{\sin\/}\nolimits\!\left(t^{2}\right)\mathrm{d}% t=\sqrt{\frac{\pi}{2}}\mathop{\mathrm{g}\/}\nolimits\!\left(\frac{a}{\sqrt{2% \pi}}\right),$ $\Re{a}>0$.

## §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).