7 Error Functions, Dawson’s and Fresnel Integrals Properties 7.1 Special Notation 7.3 Graphics

§7.2 Definitions

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Referenced by:: §12.7(ii), §7.10
Permalink:: http://dlmf.nist.gov/7.2
See also:: Annotations for Ch.7

§7.2(i) Error Functions
§7.2(ii) Dawson’s Integral
§7.2(iii) Fresnel Integrals
§7.2(iv) Auxiliary Functions
§7.2(v) Goodwin–Staton Integral

§7.2(i) Error Functions

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Keywords:: Faddeeva (or Faddeyeva) function, definition, definitions, error functions
Notes:: See Olver (1997b, pp. 43–44) and Temme (1996b, pp. 180, 182–183, 275–276).
Referenced by:: §12.13(i), §13.18(ii), §13.6(ii), §2.11(iii), §32.10(iv), §8.18(ii), §8.4
Permalink:: http://dlmf.nist.gov/7.2.i
See also:: Annotations for §7.2 and Ch.7

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 Ch.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 Ch.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, $\mathrm{i}$ : imaginary unit, $\int$ : integral and $z$ : complex variable Keywords: Faddeeva (or Faddeyeva) function, definition A&S Ref: 7.1.3 Referenced by: §7.19(i), §7.5, Erratum (V1.0.17) for Equation (7.2.3) 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 Ch.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

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Keywords:: error functions, values at infinity
See also:: Annotations for §7.2(i), §7.2 and Ch.7

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 Ch.7

§7.2(ii) Dawson’s Integral

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Keywords:: Dawson’s integral, definition
Notes:: See Olver (1997b, pp. 43–44) and Temme (1996b, pp. 180, 182–183, 275–276).
Referenced by:: §8.11(iv), §8.12, §8.4
Permalink:: http://dlmf.nist.gov/7.2.ii
See also:: Annotations for §7.2 and Ch.7

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 Ch.7

§7.2(iii) Fresnel Integrals

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Keywords:: Fresnel integrals, definition
Notes:: See Olver (1997b, pp. 43–44) and Temme (1996b, pp. 180, 182–183, 275–276).
Referenced by:: §11.10(vi)
Permalink:: http://dlmf.nist.gov/7.2.iii
See also:: Annotations for §7.2 and Ch.7

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, $\mathrm{i}$ : imaginary unit, $\int$ : 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 Ch.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 Ch.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 Ch.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

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Keywords:: Fresnel integrals, values at infinity
See also:: Annotations for §7.2(iii), §7.2 and Ch.7

7.2.9	$\displaystyle\lim_{x\to\infty}C\left(x\right)$	$\displaystyle=\tfrac{1}{2},$
7.2.9	$\displaystyle\lim_{x\to\infty}S\left(x\right)$	$\displaystyle=\tfrac{1}{2}.$
ⓘ 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 Ch.7

§7.2(iv) Auxiliary Functions

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Keywords:: Fresnel integrals, auxiliary functions, auxiliary functions for Fresnel integrals, definitions, relations to other functions
Permalink:: http://dlmf.nist.gov/7.2.iv
See also:: Annotations for §7.2 and Ch.7

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 Ch.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 Ch.7

§7.2(v) Goodwin–Staton Integral

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Keywords:: Goodwin–Staton integral, definition
Permalink:: http://dlmf.nist.gov/7.2.v
See also:: Annotations for §7.2 and Ch.7

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 Ch.7

§7.2 Definitions

Contents

§7.2(i) Error Functions

Values at Infinity

§7.2(ii) Dawson’s Integral

§7.2(iii) Fresnel Integrals

Values at Infinity

§7.2(iv) Auxiliary Functions

§7.2(v) Goodwin–Staton Integral