§7.4 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).
Keywords:: Fresnel integrals, auxiliary functions for Fresnel integrals
Permalink:: http://dlmf.nist.gov/7.4

7.4.1 $\mathop{\mathrm{erf}\/}\nolimits\!\left(-z\right)=-\mathop{\mathrm{erf}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{\mathrm{erf}\/}\nolimits z$ : error function and $z$ : complex variable
A&S Ref:: 7.1.9
Permalink:: http://dlmf.nist.gov/7.4.E1
Encodings:: TeX, pMML, png

7.4.2 $\mathop{\mathrm{erfc}\/}\nolimits\!\left(-z\right)=2-\mathop{\mathrm{erfc}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{\mathrm{erfc}\/}\nolimits z$ : complementary error function and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/7.4.E2
Encodings:: TeX, pMML, png

7.4.3 $\mathop{w\/}\nolimits\!\left(-z\right)=2e^{{-z^{2}}}-\mathop{w\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{w\/}\nolimits\!\left(z\right)$ : complementary error function, $e$ : base of exponential function and $z$ : complex variable
A&S Ref:: 7.1.11
Permalink:: http://dlmf.nist.gov/7.4.E3
Encodings:: TeX, pMML, png

7.4.4 $\mathop{F\/}\nolimits\!\left(-z\right)=-\mathop{F\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{F\/}\nolimits\!\left(z\right)$ : Dawson’s integral and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/7.4.E4
Encodings:: TeX, pMML, png

7.4.5

$\mathop{C\/}\nolimits\!\left(-z\right)=-\mathop{C\/}\nolimits\!\left(z\right),$

$\mathop{S\/}\nolimits\!\left(-z\right)=-\mathop{S\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{C\/}\nolimits\!\left(z\right)$ : Fresnel integral, $\mathop{S\/}\nolimits\!\left(z\right)$ : 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

7.4.6

$\mathop{C\/}\nolimits\!\left(iz\right)=i\mathop{C\/}\nolimits\!\left(z\right),$

$\mathop{S\/}\nolimits\!\left(iz\right)=-i\mathop{S\/}\nolimits\!\left(z\right).$

Symbols:: $\mathop{C\/}\nolimits\!\left(z\right)$ : Fresnel integral, $\mathop{S\/}\nolimits\!\left(z\right)$ : Fresnel integral 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

7.4.7

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

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

Symbols:: $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, $e$ : base of exponential function 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

7.4.8

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

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

Symbols:: $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ : auxiliary function for Fresnel integrals, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\sin\/}\nolimits z$ : 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