# §7.2 Definitions

## §7.2(i) Error Functions

 7.2.1 $\operatorname{erf}z=\frac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^{2}}\mathrm{d}t,$ ⓘ Defines: $\operatorname{erf}\NVar{z}$: error function 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 $z$: complex variable A&S Ref: 7.1.1 Referenced by: §7.5, §7.6(ii) Permalink: http://dlmf.nist.gov/7.2.E1 Encodings: TeX, pMML, png See also: Annotations for 7.2(i), 7.2 and 7
 7.2.2 $\operatorname{erfc}z=\frac{2}{\sqrt{\pi}}\int_{z}^{\infty}e^{-t^{2}}\mathrm{d}% t=1-\operatorname{erf}z,$ ⓘ Defines: $\operatorname{erfc}\NVar{z}$: complementary error function 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 $z$: complex variable A&S Ref: 7.1.2 Referenced by: §7.5, §7.7(i) Permalink: http://dlmf.nist.gov/7.2.E2 Encodings: TeX, pMML, png See also: Annotations for 7.2(i), 7.2 and 7
 7.2.3 $w\left(z\right)=e^{-z^{2}}\left(1+\frac{2i}{\sqrt{\pi}}\int_{0}^{z}e^{t^{2}}% \mathrm{d}t\right)=e^{-z^{2}}\operatorname{erfc}\left(-iz\right).$ ⓘ Defines: $w\left(\NVar{z}\right)$: Faddeeva (or Faddeyeva) function 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 of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $z$: complex variable Keywords: Faddeeva (or Faddeyeva) function A&S Ref: 7.1.3 Referenced by: §7.19(i), §7.5, Other Changes Permalink: http://dlmf.nist.gov/7.2.E3 Encodings: TeX, pMML, png Clarification (effective with 1.0.17): Originally named as a complementary error function, this has been renamed as the Faddeeva (or Faddeyeva) function. See also: Annotations for 7.2(i), 7.2 and 7

$\operatorname{erf}z$, $\operatorname{erfc}z$, and $w\left(z\right)$ are entire functions of $z$, as is $F\left(z\right)$ in the next subsection.

### Values at Infinity

 7.2.4 $\displaystyle\lim_{z\to\infty}\operatorname{erf}z$ $\displaystyle=1,$ $\displaystyle\lim_{z\to\infty}\operatorname{erfc}z$ $\displaystyle=0$, $|\operatorname{ph}z|\leq\tfrac{1}{4}\pi-\delta(<\tfrac{1}{4}\pi)$. ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\operatorname{erfc}\NVar{z}$: complementary error function, $\operatorname{erf}\NVar{z}$: error function, $\operatorname{ph}$: phase and $z$: complex variable A&S Ref: 7.1.16 (in different form) Referenced by: §7.7(i) Permalink: http://dlmf.nist.gov/7.2.E4 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 7.2(i), 7.2(i), 7.2 and 7

## §7.2(ii) Dawson’s Integral

 7.2.5 $F\left(z\right)=e^{-z^{2}}\int_{0}^{z}e^{t^{2}}\mathrm{d}t.$ ⓘ Defines: $F\left(\NVar{z}\right)$: Dawson’s integral Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral and $z$: complex variable A&S Ref: 7.1.1 Permalink: http://dlmf.nist.gov/7.2.E5 Encodings: TeX, pMML, png See also: Annotations for 7.2(ii), 7.2 and 7

## §7.2(iii) Fresnel Integrals

 7.2.6 $\displaystyle\mathcal{F}\left(z\right)$ $\displaystyle=\int_{z}^{\infty}e^{\tfrac{1}{2}\pi\mathrm{i}t^{2}}\mathrm{d}t,$ ⓘ Defines: $\mathcal{F}\left(\NVar{z}\right)$: Fresnel integral 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 $z$: complex variable Referenced by: §7.5, §7.7(ii) Permalink: http://dlmf.nist.gov/7.2.E6 Encodings: TeX, pMML, png See also: Annotations for 7.2(iii), 7.2 and 7 7.2.7 $\displaystyle C\left(z\right)$ $\displaystyle=\int_{0}^{z}\cos\left(\tfrac{1}{2}\pi t^{2}\right)\mathrm{d}t,$ ⓘ Defines: $C\left(\NVar{z}\right)$: Fresnel integral Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $z$: complex variable A&S Ref: 7.3.1 Referenced by: §7.5, §7.6(i) Permalink: http://dlmf.nist.gov/7.2.E7 Encodings: TeX, pMML, png See also: Annotations for 7.2(iii), 7.2 and 7 7.2.8 $\displaystyle S\left(z\right)$ $\displaystyle=\int_{0}^{z}\sin\left(\tfrac{1}{2}\pi t^{2}\right)\mathrm{d}t,$ ⓘ Defines: $S\left(\NVar{z}\right)$: Fresnel integral Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\sin\NVar{z}$: sine function and $z$: complex variable A&S Ref: 7.3.2 Referenced by: §7.5, §7.6(i) Permalink: http://dlmf.nist.gov/7.2.E8 Encodings: TeX, pMML, png See also: Annotations for 7.2(iii), 7.2 and 7

$\mathcal{F}\left(z\right)$, $C\left(z\right)$, and $S\left(z\right)$ are entire functions of $z$, as are $\mathrm{f}\left(z\right)$ and $\mathrm{g}\left(z\right)$ in the next subsection.

### Values at Infinity

 7.2.9 $\displaystyle\lim_{x\to\infty}C\left(x\right)$ $\displaystyle=\tfrac{1}{2},$ $\displaystyle\lim_{x\to\infty}S\left(x\right)$ $\displaystyle=\tfrac{1}{2}.$ ⓘ Symbols: $C\left(\NVar{z}\right)$: Fresnel integral, $S\left(\NVar{z}\right)$: Fresnel integral and $x$: real variable A&S Ref: 7.3.20 Referenced by: §7.5 Permalink: http://dlmf.nist.gov/7.2.E9 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 7.2(iii), 7.2(iii), 7.2 and 7

## §7.2(iv) Auxiliary Functions

 7.2.10 $\mathrm{f}\left(z\right)=\left(\tfrac{1}{2}-S\left(z\right)\right)\cos\left(% \tfrac{1}{2}\pi z^{2}\right)-\left(\tfrac{1}{2}-C\left(z\right)\right)\sin% \left(\tfrac{1}{2}\pi z^{2}\right),$ ⓘ Defines: $\mathrm{f}\left(\NVar{z}\right)$: auxiliary function for Fresnel integrals Symbols: $C\left(\NVar{z}\right)$: Fresnel integral, $S\left(\NVar{z}\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function, $\sin\NVar{z}$: sine function and $z$: complex variable A&S Ref: 7.3.5 Referenced by: §7.4, §7.5 Permalink: http://dlmf.nist.gov/7.2.E10 Encodings: TeX, pMML, png See also: Annotations for 7.2(iv), 7.2 and 7
 7.2.11 $\mathrm{g}\left(z\right)=\left(\tfrac{1}{2}-C\left(z\right)\right)\cos\left(% \tfrac{1}{2}\pi z^{2}\right)+\left(\tfrac{1}{2}-S\left(z\right)\right)\sin% \left(\tfrac{1}{2}\pi z^{2}\right).$ ⓘ Defines: $\mathrm{g}\left(\NVar{z}\right)$: auxiliary function for Fresnel integrals Symbols: $C\left(\NVar{z}\right)$: Fresnel integral, $S\left(\NVar{z}\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function, $\sin\NVar{z}$: sine function and $z$: complex variable A&S Ref: 7.3.6 Referenced by: §7.4, §7.5 Permalink: http://dlmf.nist.gov/7.2.E11 Encodings: TeX, pMML, png See also: Annotations for 7.2(iv), 7.2 and 7

## §7.2(v) Goodwin–Staton Integral

 7.2.12 $G\left(z\right)=\int_{0}^{\infty}\frac{e^{-t^{2}}}{t+z}\mathrm{d}t,$ $|\operatorname{ph}z|<\pi$. ⓘ Defines: $G\left(\NVar{z}\right)$: Goodwin–Staton integral 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, $\operatorname{ph}$: phase and $z$: complex variable A&S Ref: 27.6 (in different notation) Permalink: http://dlmf.nist.gov/7.2.E12 Encodings: TeX, pMML, png See also: Annotations for 7.2(v), 7.2 and 7