Error Functions, Dawson’s and Fresnel Integrals - 7.4 Symmetry

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
7.4.E1	${\displaystyle{\displaystyle\operatorname{erf}\left(-z\right)=-\operatorname{erf}\left(z\right)}}$ \erf@{-z} = -\erf@{z}		erf(- z) = - erf(z)	Erf[- z] == - Erf[z]	Successful	Successful	-	Successful [Tested: 7]
7.4.E2	${\displaystyle{\displaystyle\operatorname{erfc}\left(-z\right)=2-\operatorname{erfc}\left(z\right)}}$ \erfc@{-z} = 2-\erfc@{z}		erfc(- z) = 2 - erfc(z)	Erfc[- z] == 2 - Erfc[z]	Successful	Successful	-	Successful [Tested: 7]
7.4.E4	${\displaystyle{\displaystyle F\left(-z\right)=-F\left(z\right)}}$ \DawsonsintF@{-z} = -\DawsonsintF@{z}		dawson(- z) = - dawson(z)	DawsonF[- z] == - DawsonF[z]	Successful	Successful	-	Successful [Tested: 7]
7.4#Ex1	${\displaystyle{\displaystyle C\left(-z\right)=-C\left(z\right)}}$ \Fresnelcosint@{-z} = -\Fresnelcosint@{z}		FresnelC(- z) = - FresnelC(z)	FresnelC[- z] == - FresnelC[z]	Successful	Successful	-	Successful [Tested: 7]
7.4#Ex2	${\displaystyle{\displaystyle S\left(-z\right)=-S\left(z\right)}}$ \Fresnelsinint@{-z} = -\Fresnelsinint@{z}		FresnelS(- z) = - FresnelS(z)	FresnelS[- z] == - FresnelS[z]	Successful	Successful	-	Successful [Tested: 7]
7.4#Ex3	${\displaystyle{\displaystyle C\left(iz\right)=iC\left(z\right)}}$ \Fresnelcosint@{iz} = i\Fresnelcosint@{z}		FresnelC(I*z) = I*FresnelC(z)	FresnelC[I*z] == I*FresnelC[z]	Successful	Successful	-	Successful [Tested: 7]
7.4#Ex4	${\displaystyle{\displaystyle S\left(iz\right)=-iS\left(z\right)}}$ \Fresnelsinint@{iz} = -i\Fresnelsinint@{z}		FresnelS(I*z) = - I*FresnelS(z)	FresnelS[I*z] == - I*FresnelS[z]	Successful	Successful	-	Successful [Tested: 7]
7.4#Ex5	${\displaystyle{\displaystyle\mathrm{f}\left(iz\right)=(1/\sqrt{2})e^{\frac{1}{4}\pi i-\frac{1}{2}\pi iz^{2}}-i\mathrm{f}\left(z\right)}}$ \auxFresnelf@{iz} = (1/\sqrt{2})e^{\frac{1}{4}\pi i-\frac{1}{2}\pi iz^{2}}-i\auxFresnelf@{z}		Fresnelf(I*z) = (1/(sqrt(2)))*exp((1)/(4)*Pi*I -(1)/(2)*Pi*I*(z)^(2))- I*Fresnelf(z)	FresnelF[I*z] == (1/(Sqrt[2]))*Exp[Divide[1,4]*Pi*I -Divide[1,2]*Pi*I*(z)^(2)]- I*FresnelF[z]	Failure	Failure	Successful [Tested: 7]	Successful [Tested: 7]
7.4#Ex6	${\displaystyle{\displaystyle\mathrm{g}\left(iz\right)=(1/\sqrt{2})e^{-\frac{1}{4}\pi i-\frac{1}{2}\pi iz^{2}}+i\mathrm{g}\left(z\right)}}$ \auxFresnelg@{iz} = (1/\sqrt{2})e^{-\frac{1}{4}\pi i-\frac{1}{2}\pi iz^{2}}+i\auxFresnelg@{z}		Fresnelg(I*z) = (1/(sqrt(2)))*exp(-(1)/(4)*Pi*I -(1)/(2)*Pi*I*(z)^(2))+ I*Fresnelg(z)	FresnelG[I*z] == (1/(Sqrt[2]))*Exp[-Divide[1,4]*Pi*I -Divide[1,2]*Pi*I*(z)^(2)]+ I*FresnelG[z]	Failure	Failure	Successful [Tested: 7]	Successful [Tested: 7]
7.4#Ex7	${\displaystyle{\displaystyle\mathrm{f}\left(-z\right)=\sqrt{2}\cos\left(\tfrac{1}{4}\pi+\tfrac{1}{2}\pi z^{2}\right)-\mathrm{f}\left(z\right)}}$ \auxFresnelf@{-z} = \sqrt{2}\cos@{\tfrac{1}{4}\pi+\tfrac{1}{2}\pi z^{2}}-\auxFresnelf@{z}		Fresnelf(- z) = sqrt(2)*cos((1)/(4)*Pi +(1)/(2)*Pi*(z)^(2))- Fresnelf(z)	FresnelF[- z] == Sqrt[2]*Cos[Divide[1,4]*Pi +Divide[1,2]*Pi*(z)^(2)]- FresnelF[z]	Successful	Successful	-	Successful [Tested: 7]
7.4#Ex8	${\displaystyle{\displaystyle\mathrm{g}\left(-z\right)=\sqrt{2}\sin\left(\tfrac{1}{4}\pi+\tfrac{1}{2}\pi z^{2}\right)-\mathrm{g}\left(z\right)}}$ \auxFresnelg@{-z} = \sqrt{2}\sin@{\tfrac{1}{4}\pi+\tfrac{1}{2}\pi z^{2}}-\auxFresnelg@{z}		Fresnelg(- z) = sqrt(2)*sin((1)/(4)*Pi +(1)/(2)*Pi*(z)^(2))- Fresnelg(z)	FresnelG[- z] == Sqrt[2]*Sin[Divide[1,4]*Pi +Divide[1,2]*Pi*(z)^(2)]- FresnelG[z]	Successful	Failure	-	Successful [Tested: 7]