# §7.2 Definitions

## §7.2(i) Error Functions

 7.2.1 $\mathop{\mathrm{erf}\/}\nolimits z=\frac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^{2}}% \mathrm{d}t,$ Defines: $\mathop{\mathrm{erf}\/}\nolimits\NVar{z}$: 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 exponential function, $\int$: 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)
 7.2.2 $\mathop{\mathrm{erfc}\/}\nolimits z=\frac{2}{\sqrt{\pi}}\int_{z}^{\infty}e^{-t% ^{2}}\mathrm{d}t=1-\mathop{\mathrm{erf}\/}\nolimits z,$ Defines: $\mathop{\mathrm{erfc}\/}\nolimits\NVar{z}$: complementary error function Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential of $x$, $\mathop{\mathrm{erf}\/}\nolimits\NVar{z}$: error function, $\mathrm{e}$: base of exponential function, $\int$: 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)
 7.2.3 $\mathop{w\/}\nolimits\!\left(z\right)=e^{-z^{2}}\left(1+\frac{2i}{\sqrt{\pi}}% \int_{0}^{z}e^{t^{2}}\mathrm{d}t\right)=e^{-z^{2}}\mathop{\mathrm{erfc}\/}% \nolimits\!\left(-iz\right).$ Defines: $\mathop{w\/}\nolimits\!\left(\NVar{z}\right)$: complementary error function Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\mathrm{erfc}\/}\nolimits\NVar{z}$: complementary error function, $\mathrm{d}\NVar{x}$: differential of $x$, $\mathrm{e}$: base of exponential function, $\int$: integral and $z$: complex variable A&S Ref: 7.1.3 Referenced by: §7.19(i), §7.5 Permalink: http://dlmf.nist.gov/7.2.E3 Encodings: TeX, pMML, png See also: Annotations for 7.2(i)

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

### Values at Infinity

 7.2.4 $\displaystyle\lim_{z\to\infty}\mathop{\mathrm{erf}\/}\nolimits z$ $\displaystyle=1,$ $\displaystyle\lim_{z\to\infty}\mathop{\mathrm{erfc}\/}\nolimits z$ $\displaystyle=0$, $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{1}{4}\pi-\delta(<\tfrac{1}{4}\pi)$. Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\mathrm{erfc}\/}\nolimits\NVar{z}$: complementary error function, $\mathop{\mathrm{erf}\/}\nolimits\NVar{z}$: error function, $\mathop{\mathrm{ph}\/}\nolimits$: 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(ii) Dawson’s Integral

 7.2.5 $\mathop{F\/}\nolimits\!\left(z\right)=e^{-z^{2}}\int_{0}^{z}e^{t^{2}}\mathrm{d% }t.$ Defines: $\mathop{F\/}\nolimits\!\left(\NVar{z}\right)$: Dawson’s integral Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\mathrm{e}$: base of exponential function, $\int$: 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(iii) Fresnel Integrals

 7.2.6 $\displaystyle\mathop{\mathcal{F}\/}\nolimits\!\left(z\right)$ $\displaystyle=\int_{z}^{\infty}e^{\frac{1}{2}\pi it^{2}}\mathrm{d}t,$ Defines: $\mathop{\mathcal{F}\/}\nolimits\!\left(\NVar{z}\right)$: 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 exponential function, $\int$: 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) 7.2.7 $\displaystyle\mathop{C\/}\nolimits\!\left(z\right)$ $\displaystyle=\int_{0}^{z}\mathop{\cos\/}\nolimits\!\left(\tfrac{1}{2}\pi t^{2% }\right)\mathrm{d}t,$ Defines: $\mathop{C\/}\nolimits\!\left(\NVar{z}\right)$: Fresnel integral Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\cos\/}\nolimits\NVar{z}$: 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) 7.2.8 $\displaystyle\mathop{S\/}\nolimits\!\left(z\right)$ $\displaystyle=\int_{0}^{z}\mathop{\sin\/}\nolimits\!\left(\tfrac{1}{2}\pi t^{2% }\right)\mathrm{d}t,$ Defines: $\mathop{S\/}\nolimits\!\left(\NVar{z}\right)$: Fresnel integral Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\mathop{\sin\/}\nolimits\NVar{z}$: 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)

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

### Values at Infinity

 7.2.9 $\displaystyle\lim_{x\to\infty}\mathop{C\/}\nolimits\!\left(x\right)$ $\displaystyle=\tfrac{1}{2},$ $\displaystyle\lim_{x\to\infty}\mathop{S\/}\nolimits\!\left(x\right)$ $\displaystyle=\tfrac{1}{2}.$ Symbols: $\mathop{C\/}\nolimits\!\left(\NVar{z}\right)$: Fresnel integral, $\mathop{S\/}\nolimits\!\left(\NVar{z}\right)$: 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(iv) Auxiliary Functions

 7.2.10 $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)=\left(\tfrac{1}{2}-\mathop{S\/}% \nolimits\!\left(z\right)\right)\mathop{\cos\/}\nolimits\!\left(\tfrac{1}{2}% \pi z^{2}\right)-\left(\tfrac{1}{2}-\mathop{C\/}\nolimits\!\left(z\right)% \right)\mathop{\sin\/}\nolimits\!\left(\tfrac{1}{2}\pi z^{2}\right),$ Defines: $\mathop{\mathrm{f}\/}\nolimits\!\left(\NVar{z}\right)$: auxiliary function for Fresnel integrals Symbols: $\mathop{C\/}\nolimits\!\left(\NVar{z}\right)$: Fresnel integral, $\mathop{S\/}\nolimits\!\left(\NVar{z}\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $\mathop{\sin\/}\nolimits\NVar{z}$: 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)
 7.2.11 $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)=\left(\tfrac{1}{2}-\mathop{C\/}% \nolimits\!\left(z\right)\right)\mathop{\cos\/}\nolimits\!\left(\tfrac{1}{2}% \pi z^{2}\right)+\left(\tfrac{1}{2}-\mathop{S\/}\nolimits\!\left(z\right)% \right)\mathop{\sin\/}\nolimits\!\left(\tfrac{1}{2}\pi z^{2}\right).$ Defines: $\mathop{\mathrm{g}\/}\nolimits\!\left(\NVar{z}\right)$: auxiliary function for Fresnel integrals Symbols: $\mathop{C\/}\nolimits\!\left(\NVar{z}\right)$: Fresnel integral, $\mathop{S\/}\nolimits\!\left(\NVar{z}\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $\mathop{\sin\/}\nolimits\NVar{z}$: 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(v) Goodwin–Staton Integral

 7.2.12 $\mathop{G\/}\nolimits\!\left(z\right)=\int_{0}^{\infty}\frac{e^{-t^{2}}}{t+z}% \mathrm{d}t,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$. Defines: $\mathop{G\/}\nolimits\!\left(\NVar{z}\right)$: 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 exponential function, $\int$: integral, $\mathop{\mathrm{ph}\/}\nolimits$: 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)