# §7.4 Symmetry

 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 $\displaystyle C\left(-z\right)$ $\displaystyle=-C\left(z\right),$ $\displaystyle S\left(-z\right)$ $\displaystyle=-S\left(z\right),$ ⓘ 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 $\displaystyle C\left(iz\right)$ $\displaystyle=iC\left(z\right),$ $\displaystyle S\left(iz\right)$ $\displaystyle=-iS\left(z\right).$ ⓘ 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 $\displaystyle\mathrm{f}\left(iz\right)$ $\displaystyle=(1/\sqrt{2})e^{\frac{1}{4}\pi i-\frac{1}{2}\pi iz^{2}}-i\mathrm{% f}\left(z\right),$ $\displaystyle\mathrm{g}\left(iz\right)$ $\displaystyle=(1/\sqrt{2})e^{-\frac{1}{4}\pi i-\frac{1}{2}\pi iz^{2}}+i\mathrm% {g}\left(z\right).$
 7.4.8 $\displaystyle\mathrm{f}\left(-z\right)$ $\displaystyle=\sqrt{2}\cos\left(\tfrac{1}{4}\pi+\tfrac{1}{2}\pi z^{2}\right)-% \mathrm{f}\left(z\right),$ $\displaystyle\mathrm{g}\left(-z\right)$ $\displaystyle=\sqrt{2}\sin\left(\tfrac{1}{4}\pi+\tfrac{1}{2}\pi z^{2}\right)-% \mathrm{g}\left(z\right).$