7.2

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
7.2.E1	${\displaystyle{\displaystyle\operatorname{erf}z=\frac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^{2}}\mathrm{d}t}}$ \erf@@{z} = \frac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^{2}}\diff{t}		erf(z) = (2)/(sqrt(Pi))*int(exp(- (t)^(2)), t = 0..z)	Erf[z] == Divide[2,Sqrt[Pi]]*Integrate[Exp[- (t)^(2)], {t, 0, z}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 7]
7.2.E2	${\displaystyle{\displaystyle\operatorname{erfc}z=\frac{2}{\sqrt{\pi}}\int_{z}^{\infty}e^{-t^{2}}\mathrm{d}t}}$ \erfc@@{z} = \frac{2}{\sqrt{\pi}}\int_{z}^{\infty}e^{-t^{2}}\diff{t}		erfc(z) = (2)/(sqrt(Pi))*int(exp(- (t)^(2)), t = z..infinity)	Erfc[z] == Divide[2,Sqrt[Pi]]*Integrate[Exp[- (t)^(2)], {t, z, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 7]
7.2.E2	${\displaystyle{\displaystyle\frac{2}{\sqrt{\pi}}\int_{z}^{\infty}e^{-t^{2}}\mathrm{d}t=1-\operatorname{erf}z}}$ \frac{2}{\sqrt{\pi}}\int_{z}^{\infty}e^{-t^{2}}\diff{t} = 1-\erf@@{z}		(2)/(sqrt(Pi))*int(exp(- (t)^(2)), t = z..infinity) = 1 - erf(z)	Divide[2,Sqrt[Pi]]*Integrate[Exp[- (t)^(2)], {t, z, Infinity}, GenerateConditions->None] == 1 - Erf[z]	Successful	Successful	-	Successful [Tested: 7]
7.2.E3	${\displaystyle{\displaystyle e^{-z^{2}}\left(1+\frac{2i}{\sqrt{\pi}}\int_{0}^{z}e^{t^{2}}\mathrm{d}t\right)=e^{-z^{2}}\operatorname{erfc}\left(-iz\right)}}$ e^{-z^{2}}\left(1+\frac{2i}{\sqrt{\pi}}\int_{0}^{z}e^{t^{2}}\diff{t}\right) = e^{-z^{2}}\erfc@{-iz}		exp(- (z)^(2))*(1 +(2*I)/(sqrt(Pi))*int(exp((t)^(2)), t = 0..z)) = exp(- (z)^(2))*erfc(- I*z)	Exp[- (z)^(2)]*(1 +Divide[2*I,Sqrt[Pi]]*Integrate[Exp[(t)^(2)], {t, 0, z}, GenerateConditions->None]) == Exp[- (z)^(2)]*Erfc[- I*z]	Successful	Successful	-	Successful [Tested: 7]
7.2#Ex1	${\displaystyle{\displaystyle\lim_{z\to\infty}\operatorname{erf}z=1}}$ \lim_{z\to\infty}\erf@@{z} = 1		limit(erf(z), z = infinity) = 1	Limit[Erf[z], z -> Infinity, GenerateConditions->None] == 1	Successful	Successful	-	Successful [Tested: 1]
7.2#Ex2	${\displaystyle{\displaystyle\lim_{z\to\infty}\operatorname{erfc}z=0}}$ \lim_{z\to\infty}\erfc@@{z} = 0	${\displaystyle{\displaystyle|\operatorname{ph}z|\leq\tfrac{1}{4}\pi-\delta(<\tfrac{1}{4}\pi)}}$	limit(erfc(z), z = infinity) = 0	Limit[Erfc[z], z -> Infinity, GenerateConditions->None] == 0	Successful	Successful	-	Successful [Tested: 1]
7.2.E5	${\displaystyle{\displaystyle F\left(z\right)=e^{-z^{2}}\int_{0}^{z}e^{t^{2}}\mathrm{d}t}}$ \DawsonsintF@{z} = e^{-z^{2}}\int_{0}^{z}e^{t^{2}}\diff{t}		dawson(z) = exp(- (z)^(2))*int(exp((t)^(2)), t = 0..z)	DawsonF[z] == Exp[- (z)^(2)]*Integrate[Exp[(t)^(2)], {t, 0, z}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 7]
7.2.E6	${\displaystyle{\displaystyle\mathcal{F}\left(z\right)=\int_{z}^{\infty}e^{\tfrac{1}{2}\pi\mathrm{i}t^{2}}\mathrm{d}t}}$ \FresnelintF@{z} = \int_{z}^{\infty}e^{\tfrac{1}{2}\pi\iunit t^{2}}\diff{t}		Error	(1+I)/2-FresnelC[z]-I*FresnelS[z] == Integrate[Exp[Divide[1,2]*Pi*I*(t)^(2)], {t, z, Infinity}, GenerateConditions->None]	Missing Macro Error	Failure	-	Failed [2 / 7] Result: Complex[-0.17236809983536389, -1.1316008349021112] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} Result: Complex[0.17236809983536283, 1.1316008349021118] Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}
7.2.E7	${\displaystyle{\displaystyle C\left(z\right)=\int_{0}^{z}\cos\left(\tfrac{1}{2}\pi t^{2}\right)\mathrm{d}t}}$ \Fresnelcosint@{z} = \int_{0}^{z}\cos@{\tfrac{1}{2}\pi t^{2}}\diff{t}		FresnelC(z) = int(cos((1)/(2)*Pi*(t)^(2)), t = 0..z)	FresnelC[z] == Integrate[Cos[Divide[1,2]*Pi*(t)^(2)], {t, 0, z}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 7]
7.2.E8	${\displaystyle{\displaystyle S\left(z\right)=\int_{0}^{z}\sin\left(\tfrac{1}{2}\pi t^{2}\right)\mathrm{d}t}}$ \Fresnelsinint@{z} = \int_{0}^{z}\sin@{\tfrac{1}{2}\pi t^{2}}\diff{t}		FresnelS(z) = int(sin((1)/(2)*Pi*(t)^(2)), t = 0..z)	FresnelS[z] == Integrate[Sin[Divide[1,2]*Pi*(t)^(2)], {t, 0, z}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 7]
7.2#Ex3	${\displaystyle{\displaystyle\lim_{x\to\infty}C\left(x\right)=\tfrac{1}{2}}}$ \lim_{x\to\infty}\Fresnelcosint@{x} = \tfrac{1}{2}		limit(FresnelC(x), x = infinity) = (1)/(2)	Limit[FresnelC[x], x -> Infinity, GenerateConditions->None] == Divide[1,2]	Successful	Successful	-	Successful [Tested: 1]
7.2#Ex4	${\displaystyle{\displaystyle\lim_{x\to\infty}S\left(x\right)=\tfrac{1}{2}}}$ \lim_{x\to\infty}\Fresnelsinint@{x} = \tfrac{1}{2}		limit(FresnelS(x), x = infinity) = (1)/(2)	Limit[FresnelS[x], x -> Infinity, GenerateConditions->None] == Divide[1,2]	Successful	Successful	-	Successful [Tested: 1]
7.2.E10	${\displaystyle{\displaystyle\mathrm{f}\left(z\right)=\left(\tfrac{1}{2}-S\left(z\right)\right)\cos\left(\tfrac{1}{2}\pi z^{2}\right)-\left(\tfrac{1}{2}-C\left(z\right)\right)\sin\left(\tfrac{1}{2}\pi z^{2}\right)}}$ \auxFresnelf@{z} = \left(\tfrac{1}{2}-\Fresnelsinint@{z}\right)\cos@{\tfrac{1}{2}\pi z^{2}}-\left(\tfrac{1}{2}-\Fresnelcosint@{z}\right)\sin@{\tfrac{1}{2}\pi z^{2}}		Fresnelf(z) = ((1)/(2)- FresnelS(z))*cos((1)/(2)*Pi*(z)^(2))-((1)/(2)- FresnelC(z))*sin((1)/(2)*Pi*(z)^(2))	FresnelF[z] == (Divide[1,2]- FresnelS[z])*Cos[Divide[1,2]*Pi*(z)^(2)]-(Divide[1,2]- FresnelC[z])*Sin[Divide[1,2]*Pi*(z)^(2)]	Successful	Successful	-	Successful [Tested: 7]
7.2.E11	${\displaystyle{\displaystyle\mathrm{g}\left(z\right)=\left(\tfrac{1}{2}-C\left(z\right)\right)\cos\left(\tfrac{1}{2}\pi z^{2}\right)+\left(\tfrac{1}{2}-S\left(z\right)\right)\sin\left(\tfrac{1}{2}\pi z^{2}\right)}}$ \auxFresnelg@{z} = \left(\tfrac{1}{2}-\Fresnelcosint@{z}\right)\cos@{\tfrac{1}{2}\pi z^{2}}+\left(\tfrac{1}{2}-\Fresnelsinint@{z}\right)\sin@{\tfrac{1}{2}\pi z^{2}}		Fresnelg(z) = ((1)/(2)- FresnelC(z))*cos((1)/(2)*Pi*(z)^(2))+((1)/(2)- FresnelS(z))*sin((1)/(2)*Pi*(z)^(2))	FresnelG[z] == (Divide[1,2]- FresnelC[z])*Cos[Divide[1,2]*Pi*(z)^(2)]+(Divide[1,2]- FresnelS[z])*Sin[Divide[1,2]*Pi*(z)^(2)]	Successful	Successful	-	Successful [Tested: 7]