Content-Length: 84379 | pFad | https://dlmf.nist.gov/./././././././././.././bib/.././././bib/.././.././../././../././7.4#info

DLMF: §7.4 Symmetry ‣ Properties ‣ Chapter 7 Error Functions, Dawson’s and Fresnel Integrals

About the Project

7 Error Functions, Dawson’s and Fresnel Integrals Properties 7.3 Graphics 7.5 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		$\displaystyle\operatorname{erf}\left(-z\right)$	$\displaystyle=-\operatorname{erf}\left(z\right),$
ⓘ Symbols: $\operatorname{erf}\NVar{z}$ : 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		$\displaystyle\operatorname{erfc}\left(-z\right)$	$\displaystyle=2-\operatorname{erfc}\left(z\right),$
ⓘ Symbols: $\operatorname{erfc}\NVar{z}$ : 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		$\displaystyle w\left(-z\right)$	$\displaystyle=2e^{-z^{2}}-w\left(z\right).$
ⓘ Symbols: $w\left(\NVar{z}\right)$ : 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		$\displaystyle F\left(-z\right)$	$\displaystyle=-F\left(z\right).$
ⓘ Symbols: $F\left(\NVar{z}\right)$ : 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		$\displaystyle C\left(-z\right)$	$\displaystyle=-C\left(z\right),$
7.4.5		$\displaystyle S\left(-z\right)$	$\displaystyle=-S\left(z\right),$
ⓘ Symbols: $C\left(\NVar{z}\right)$ : Fresnel integral, $S\left(\NVar{z}\right)$ : 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		$\displaystyle C\left(iz\right)$	$\displaystyle=iC\left(z\right),$
7.4.6		$\displaystyle S\left(iz\right)$	$\displaystyle=-iS\left(z\right).$
ⓘ Symbols: $C\left(\NVar{z}\right)$ : Fresnel integral, $S\left(\NVar{z}\right)$ : 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		$\displaystyle\mathrm{f}\left(iz\right)$	$\displaystyle=(1/\sqrt{2})e^{\frac{1}{4}\pi i-\frac{1}{2}\pi iz^{2}}-i\mathrm{% f}\left(z\right),$
7.4.7		$\displaystyle\mathrm{g}\left(iz\right)$	$\displaystyle=(1/\sqrt{2})e^{-\frac{1}{4}\pi i-\frac{1}{2}\pi iz^{2}}+i\mathrm% {g}\left(z\right).$
ⓘ Symbols: $\mathrm{f}\left(\NVar{z}\right)$ : auxiliary function for Fresnel integrals, $\mathrm{g}\left(\NVar{z}\right)$ : 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		$\displaystyle\mathrm{f}\left(-z\right)$	$\displaystyle=\sqrt{2}\cos\left(\tfrac{1}{4}\pi+\tfrac{1}{2}\pi z^{2}\right)-% \mathrm{f}\left(z\right),$
7.4.8		$\displaystyle\mathrm{g}\left(-z\right)$	$\displaystyle=\sqrt{2}\sin\left(\tfrac{1}{4}\pi+\tfrac{1}{2}\pi z^{2}\right)-% \mathrm{g}\left(z\right).$
ⓘ Symbols: $\mathrm{f}\left(\NVar{z}\right)$ : auxiliary function for Fresnel integrals, $\mathrm{g}\left(\NVar{z}\right)$ : auxiliary function for Fresnel integrals, $\pi$ : 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 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 Graphics 7.5 Interrelations

© 2010–2024 NIST / Disclaimer / Feedback; Version 1.2.3; Release date 2024-12-15.

NIST

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov