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 $\operatorname{erf}\left(-z\right)=-\operatorname{erf}\left(z\right),$

ⓘ

Symbols:: $\operatorname{erf}\NVar{z}$ : error function and : 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 $\operatorname{erfc}\left(-z\right)=2-\operatorname{erfc}\left(z\right),$

ⓘ

Symbols:: $\operatorname{erfc}\NVar{z}$ : complementary error function and : 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 $w\left(-z\right)=2e^{-z^{2}}-w\left(z\right).$

ⓘ

Symbols:: $w\left(\NVar{z}\right)$ : Faddeeva (or Faddeyeva) function, $\mathrm{e}$ : base of natural logarithm and : 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:: $F\left(\NVar{z}\right)$ : Dawson’s integral and : 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:: $C\left(\NVar{z}\right)$ : Fresnel integral, $S\left(\NVar{z}\right)$ : Fresnel integral and : 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

$C\left(iz\right)=iC\left(z\right),$

$S\left(iz\right)=-iS\left(z\right).$

ⓘ

Symbols:: $C\left(\NVar{z}\right)$ : Fresnel integral, $S\left(\NVar{z}\right)$ : Fresnel integral, $\mathrm{i}$ : imaginary unit and : 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

$\mathrm{f}\left(iz\right)=(1/\sqrt{2})e^{\frac{1}{4}\pi i-\frac{1}{2}\pi iz^{2% }}-i\mathrm{f}\left(z\right),$

$\mathrm{g}\left(iz\right)=(1/\sqrt{2})e^{-\frac{1}{4}\pi i-\frac{1}{2}\pi iz^{% 2}}+i\mathrm{g}\left(z\right).$

ⓘ

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 : 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

$\mathrm{f}\left(-z\right)=\sqrt{2}\cos\left(\tfrac{1}{4}\pi+\tfrac{1}{2}\pi z^% {2}\right)-\mathrm{f}\left(z\right),$

$\mathrm{g}\left(-z\right)=\sqrt{2}\sin\left(\tfrac{1}{4}\pi+\tfrac{1}{2}\pi z^% {2}\right)-\mathrm{g}\left(z\right).$

ⓘ

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 : 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

© 2010–2026 NIST / Disclaimer / Feedback; Version 1.2.7; Release date 2026-06-15 Math-Image version (See MathML) .

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