Exponential, Logarithmic, Sine, and Cosine Integrals - 7.2 Definitions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
7.2.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \erf@@{z} = \frac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^{2}}\diff{t}}
\erf@@{z} = \frac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^{2}}\diff{t}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \erfc@@{z} = \frac{2}{\sqrt{\pi}}\int_{z}^{\infty}e^{-t^{2}}\diff{t}}
\erfc@@{z} = \frac{2}{\sqrt{\pi}}\int_{z}^{\infty}e^{-t^{2}}\diff{t}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{2}{\sqrt{\pi}}\int_{z}^{\infty}e^{-t^{2}}\diff{t} = 1-\erf@@{z}}
\frac{2}{\sqrt{\pi}}\int_{z}^{\infty}e^{-t^{2}}\diff{t} = 1-\erf@@{z}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
(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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle e^{-z^{2}}\left(1+\frac{2i}{\sqrt{\pi}}\int_{0}^{z}e^{t^{2}}\diff{t}\right) = e^{-z^{2}}\erfc@{-iz}}
e^{-z^{2}}\left(1+\frac{2i}{\sqrt{\pi}}\int_{0}^{z}e^{t^{2}}\diff{t}\right) = e^{-z^{2}}\erfc@{-iz}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lim_{z\to\infty}\erf@@{z} = 1}
\lim_{z\to\infty}\erf@@{z} = 1		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
limit(erf(z), z = infinity) = 1

Limit[Erf[z], z -> Infinity, GenerateConditions->None] == 1
Successful Successful - Successful [Tested: 1]
7.2#Ex2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lim_{z\to\infty}\erfc@@{z} = 0}
\lim_{z\to\infty}\erfc@@{z} = 0		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\phase@@{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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \DawsonsintF@{z} = e^{-z^{2}}\int_{0}^{z}e^{t^{2}}\diff{t}}
\DawsonsintF@{z} = e^{-z^{2}}\int_{0}^{z}e^{t^{2}}\diff{t}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \FresnelintF@{z} = \int_{z}^{\infty}e^{\tfrac{1}{2}\pi\iunit t^{2}}\diff{t}}
\FresnelintF@{z} = \int_{z}^{\infty}e^{\tfrac{1}{2}\pi\iunit t^{2}}\diff{t}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Fresnelcosint@{z} = \int_{0}^{z}\cos@{\tfrac{1}{2}\pi t^{2}}\diff{t}}
\Fresnelcosint@{z} = \int_{0}^{z}\cos@{\tfrac{1}{2}\pi t^{2}}\diff{t}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Fresnelsinint@{z} = \int_{0}^{z}\sin@{\tfrac{1}{2}\pi t^{2}}\diff{t}}
\Fresnelsinint@{z} = \int_{0}^{z}\sin@{\tfrac{1}{2}\pi t^{2}}\diff{t}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lim_{x\to\infty}\Fresnelcosint@{x} = \tfrac{1}{2}}
\lim_{x\to\infty}\Fresnelcosint@{x} = \tfrac{1}{2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lim_{x\to\infty}\Fresnelsinint@{x} = \tfrac{1}{2}}
\lim_{x\to\infty}\Fresnelsinint@{x} = \tfrac{1}{2}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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}}}
\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}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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}}}
\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}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } 
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]
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