7 Error Functions, Dawson’s and Fresnel IntegralsProperties7.3 Graphics7.5 Interrelations

§7.4 Symmetry

ⓘ

Keywords:

Fresnel integrals, auxiliary functions for Fresnel integrals, symmetry

Notes:

(7.4.7) follows from (7.2.10), (7.2.11), and (7.4.6). (7.4.8) follows from (7.4.7).

Permalink:

http://dlmf.nist.gov/7.4

See also:

Annotations for Ch.7

7.4.1		$\mathrm{erf}\left(-z\right)$	$=-\mathrm{erf}\left(z\right),$
ⓘ Symbols: $\mathrm{erf}z$: error function and $z$: complex variable A&S Ref: 7.1.9 Permalink: http://dlmf.nist.gov/7.4.E1 Encodings: TeX, pMML, png See also: Annotations for §7.4 and Ch.7
7.4.2		$\mathrm{erfc}\left(-z\right)$	$=2-\mathrm{erfc}\left(z\right),$
ⓘ Symbols: $\mathrm{erfc}z$: complementary error function and $z$: complex variable Permalink: http://dlmf.nist.gov/7.4.E2 Encodings: TeX, pMML, png See also: Annotations for §7.4 and Ch.7
7.4.3		$w\left(-z\right)$	$=2{\mathrm{e}}^{-z^2}-w\left(z\right).$
ⓘ Symbols: $w\left(z\right)$: Faddeeva (or Faddeyeva) function, $\mathrm{e}$: base of natural logarithm and $z$: complex variable A&S Ref: 7.1.11 Permalink: http://dlmf.nist.gov/7.4.E3 Encodings: TeX, pMML, png See also: Annotations for §7.4 and Ch.7
7.4.4		$F\left(-z\right)$	$=-F\left(z\right).$
ⓘ Symbols: $F\left(z\right)$: Dawson’s integral and $z$: complex variable Permalink: http://dlmf.nist.gov/7.4.E4 Encodings: TeX, pMML, png See also: Annotations for §7.4 and Ch.7

7.4.5		$C\left(-z\right)$	$=-C\left(z\right),$
		$S\left(-z\right)$	$=-S\left(z\right),$
ⓘ Symbols: $C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral and $z$: complex variable A&S Ref: 7.3.17 Permalink: http://dlmf.nist.gov/7.4.E5 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §7.4 and Ch.7

7.4.6		$C\left(\mathrm{i}z\right)$	$=\mathrm{i}C\left(z\right),$
		$S\left(\mathrm{i}z\right)$	$=-\mathrm{i}S\left(z\right).$
ⓘ Symbols: $C\left(z\right)$: Fresnel integral, $S\left(z\right)$: Fresnel integral, $\mathrm{i}$: imaginary unit and $z$: complex variable A&S Ref: 7.3.18 Referenced by: §7.4 Permalink: http://dlmf.nist.gov/7.4.E6 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §7.4 and Ch.7

7.4.7		$\mathrm{f}\left(\mathrm{i}z\right)$	$=(1/\sqrt{2}){\mathrm{e}}^{\frac{1}{4}\pi \mathrm{i}-\frac{1}{2}\pi \mathrm{i}{z}^2}-\mathrm{i}\mathrm{f}\left(z\right),$
		$\mathrm{g}\left(\mathrm{i}z\right)$	$=(1/\sqrt{2}){\mathrm{e}}^{-\frac{1}{4}\pi \mathrm{i}-\frac{1}{2}\pi \mathrm{i}{z}^2}+\mathrm{i}\mathrm{g}\left(z\right).$
ⓘ Symbols: $\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable Referenced by: §7.12(ii), §7.4 Permalink: http://dlmf.nist.gov/7.4.E7 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §7.4 and Ch.7

7.4.8		$\mathrm{f}\left(-z\right)$	$=\sqrt{2}\cos \left(\frac{1}{4}\pi +\frac{1}{2}\pi z^2\right)-\mathrm{f}\left(z\right),$
		$\mathrm{g}\left(-z\right)$	$=\sqrt{2}\sin \left(\frac{1}{4}\pi +\frac{1}{2}\pi z^2\right)-\mathrm{g}\left(z\right).$
ⓘ Symbols: $\mathrm{f}\left(z\right)$: auxiliary function for Fresnel integrals, $\mathrm{g}\left(z\right)$: auxiliary function for Fresnel integrals, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\sin z$: sine function and $z$: complex variable Referenced by: §7.12(ii), §7.4, §7.5 Permalink: http://dlmf.nist.gov/7.4.E8 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §7.4 and Ch.7

7.3 Graphics7.5 Interrelations

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