7 Error Functions, Dawson’s and Fresnel IntegralsProperties7.1 Special Notation7.3 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

Contents

1. §7.2(i) Error Functions
2. §7.2(ii) Dawson’s Integral
3. §7.2(iii) Fresnel Integrals
4. §7.2(iv) Auxiliary Functions
5. §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		$$\mathrm{erf}z=\frac{2}{\sqrt{\pi }}{\int }_0^z{\mathrm{e}}^{-t^2}dt,$$
ⓘ Defines: $\mathrm{erf}z$: error function Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: 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		$$\mathrm{erfc}z=\frac{2}{\sqrt{\pi }}{\int }_z^{\mathrm{\infty }}{\mathrm{e}}^{-t^2}dt=1-\mathrm{erf}z,$$
ⓘ Defines: $\mathrm{erfc}z$: complementary error function Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{erf}z$: 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		$$w\left(z\right)={\mathrm{e}}^{-z^2}\left(1+\frac{2\mathrm{i}}{\sqrt{\pi }}{\int }_0^z{\mathrm{e}}^{t^2}dt\right)={\mathrm{e}}^{-z^2}\mathrm{erfc}\left(-\mathrm{i}z\right).$$
ⓘ Defines: $w\left(z\right)$: Faddeeva (or Faddeyeva) function Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $dx$: 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

$\mathrm{erf}z$, $\mathrm{erfc}z$, 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		$\underset{z\to \mathrm{\infty }}{lim}\mathrm{erf}z$	$=1,$
		$\underset{z\to \mathrm{\infty }}{lim}\mathrm{erfc}z$	$=0$,
	.
ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{erf}z$: error function, $\mathrm{ph}$: 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		$$F\left(z\right)={\mathrm{e}}^{-z^2}{\int }_0^z{\mathrm{e}}^{t^2}dt.$$
ⓘ Defines: $F\left(z\right)$: Dawson’s integral Symbols: $dx$: 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		$ℱ\left(z\right)$	$={\displaystyle {\int }_z^{\mathrm{\infty }}}{\mathrm{e}}^{\frac{1}{2}\pi \mathrm{i}{t}^2}dt,$
ⓘ Defines: $ℱ\left(z\right)$: Fresnel integral Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: 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		$C\left(z\right)$	$={\displaystyle {\int }_0^z}\cos \left(\frac{1}{2}\pi t^2\right)dt,$
ⓘ Defines: $C\left(z\right)$: Fresnel integral Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: 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		$S\left(z\right)$	$={\displaystyle {\int }_0^z}\sin \left(\frac{1}{2}\pi t^2\right)dt,$
ⓘ Defines: $S\left(z\right)$: Fresnel integral Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\int$: integral, $\sin z$: 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

$ℱ\left(z\right)$, $C\left(z\right)$, 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		$\underset{x\to \mathrm{\infty }}{lim}C\left(x\right)$	$=\frac{1}{2},$
		$\underset{x\to \mathrm{\infty }}{lim}S\left(x\right)$	$=\frac{1}{2}.$
ⓘ Symbols: $C\left(z\right)$: Fresnel integral, $S\left(z\right)$: 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		$$\mathrm{f}\left(z\right)=\left(\frac{1}{2}-S\left(z\right)\right)\cos \left(\frac{1}{2}\pi z^2\right)-\left(\frac{1}{2}-C\left(z\right)\right)\sin \left(\frac{1}{2}\pi z^2\right),$$
ⓘ Defines: $\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals Symbols: $C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: 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		$$\mathrm{g}\left(z\right)=\left(\frac{1}{2}-C\left(z\right)\right)\cos \left(\frac{1}{2}\pi z^2\right)+\left(\frac{1}{2}-S\left(z\right)\right)\sin \left(\frac{1}{2}\pi z^2\right).$$
ⓘ Defines: $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals Symbols: $C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: 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		$$G\left(z\right)={\int }_0^{\mathrm{\infty }}\frac{{\mathrm{e}}^{-t^2}}{t+z}dt,$$
.
ⓘ Defines: $G\left(z\right)$: Goodwin–Staton integral Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: 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

7.1 Special Notation7.3 Graphics

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