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{{DISPLAYTITLE:Exponential, Logarithmic, Sine, and Cosine Integrals - 7.2 Definitions}}

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! scope="col" style="position: sticky; top: 0;" | DLMF 
! scope="col" style="position: sticky; top: 0;" | Formula 
! scope="col" style="position: sticky; top: 0;" | Constraints 
! scope="col" style="position: sticky; top: 0;" | Maple
! scope="col" style="position: sticky; top: 0;" | Mathematica
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Mathematica
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/7.2.E1 7.2.E1] || [[Item:Q2321|<math>\erf@@{z} = \frac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^{2}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\erf@@{z} = \frac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^{2}}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>erf(z) = (2)/(sqrt(Pi))*int(exp(- (t)^(2)), t = 0..z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Erf[z] == Divide[2,Sqrt[Pi]]*Integrate[Exp[- (t)^(2)], {t, 0, z}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
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| [https://dlmf.nist.gov/7.2.E2 7.2.E2] || [[Item:Q2322|<math>\erfc@@{z} = \frac{2}{\sqrt{\pi}}\int_{z}^{\infty}e^{-t^{2}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\erfc@@{z} = \frac{2}{\sqrt{\pi}}\int_{z}^{\infty}e^{-t^{2}}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>erfc(z) = (2)/(sqrt(Pi))*int(exp(- (t)^(2)), t = z..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Erfc[z] == Divide[2,Sqrt[Pi]]*Integrate[Exp[- (t)^(2)], {t, z, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
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| [https://dlmf.nist.gov/7.2.E2 7.2.E2] || [[Item:Q2322|<math>\frac{2}{\sqrt{\pi}}\int_{z}^{\infty}e^{-t^{2}}\diff{t} = 1-\erf@@{z}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\frac{2}{\sqrt{\pi}}\int_{z}^{\infty}e^{-t^{2}}\diff{t} = 1-\erf@@{z}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>(2)/(sqrt(Pi))*int(exp(- (t)^(2)), t = z..infinity) = 1 - erf(z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[2,Sqrt[Pi]]*Integrate[Exp[- (t)^(2)], {t, z, Infinity}, GenerateConditions->None] == 1 - Erf[z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
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| [https://dlmf.nist.gov/7.2.E3 7.2.E3] || [[Item:Q2323|<math>e^{-z^{2}}\left(1+\frac{2i}{\sqrt{\pi}}\int_{0}^{z}e^{t^{2}}\diff{t}\right) = e^{-z^{2}}\erfc@{-iz}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>e^{-z^{2}}\left(1+\frac{2i}{\sqrt{\pi}}\int_{0}^{z}e^{t^{2}}\diff{t}\right) = e^{-z^{2}}\erfc@{-iz}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>exp(- (z)^(2))*(1 +(2*I)/(sqrt(Pi))*int(exp((t)^(2)), t = 0..z)) = exp(- (z)^(2))*erfc(- I*z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Exp[- (z)^(2)]*(1 +Divide[2*I,Sqrt[Pi]]*Integrate[Exp[(t)^(2)], {t, 0, z}, GenerateConditions->None]) == Exp[- (z)^(2)]*Erfc[- I*z]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
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| [https://dlmf.nist.gov/7.2#Ex1 7.2#Ex1] || [[Item:Q2324|<math>\lim_{z\to\infty}\erf@@{z} = 1</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\lim_{z\to\infty}\erf@@{z} = 1</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>limit(erf(z), z = infinity) = 1</syntaxhighlight> || <syntaxhighlight lang=mathematica>Limit[Erf[z], z -> Infinity, GenerateConditions->None] == 1</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
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| [https://dlmf.nist.gov/7.2#Ex2 7.2#Ex2] || [[Item:Q2325|<math>\lim_{z\to\infty}\erfc@@{z} = 0</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\lim_{z\to\infty}\erfc@@{z} = 0</syntaxhighlight> || <math>|\phase@@{z}| \leq \tfrac{1}{4}\pi-\delta(<\tfrac{1}{4}\pi)</math> || <syntaxhighlight lang=mathematica>limit(erfc(z), z = infinity) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>Limit[Erfc[z], z -> Infinity, GenerateConditions->None] == 0</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
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| [https://dlmf.nist.gov/7.2.E5 7.2.E5] || [[Item:Q2326|<math>\DawsonsintF@{z} = e^{-z^{2}}\int_{0}^{z}e^{t^{2}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\DawsonsintF@{z} = e^{-z^{2}}\int_{0}^{z}e^{t^{2}}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>dawson(z) = exp(- (z)^(2))*int(exp((t)^(2)), t = 0..z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>DawsonF[z] == Exp[- (z)^(2)]*Integrate[Exp[(t)^(2)], {t, 0, z}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
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| [https://dlmf.nist.gov/7.2.E6 7.2.E6] || [[Item:Q2327|<math>\FresnelintF@{z} = \int_{z}^{\infty}e^{\tfrac{1}{2}\pi\iunit t^{2}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\FresnelintF@{z} = \int_{z}^{\infty}e^{\tfrac{1}{2}\pi\iunit t^{2}}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || <syntaxhighlight lang=mathematica>(1+I)/2-FresnelC[z]-I*FresnelS[z] == Integrate[Exp[Divide[1,2]*Pi*I*(t)^(2)], {t, z, Infinity}, GenerateConditions->None]</syntaxhighlight> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 7]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Complex[-0.17236809983536389, -1.1316008349021112]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Complex[0.17236809983536283, 1.1316008349021118]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[-1, 3]], Pi]]]}</syntaxhighlight><br></div></div>
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| [https://dlmf.nist.gov/7.2.E7 7.2.E7] || [[Item:Q2328|<math>\Fresnelcosint@{z} = \int_{0}^{z}\cos@{\tfrac{1}{2}\pi t^{2}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Fresnelcosint@{z} = \int_{0}^{z}\cos@{\tfrac{1}{2}\pi t^{2}}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>FresnelC(z) = int(cos((1)/(2)*Pi*(t)^(2)), t = 0..z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>FresnelC[z] == Integrate[Cos[Divide[1,2]*Pi*(t)^(2)], {t, 0, z}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
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| [https://dlmf.nist.gov/7.2.E8 7.2.E8] || [[Item:Q2329|<math>\Fresnelsinint@{z} = \int_{0}^{z}\sin@{\tfrac{1}{2}\pi t^{2}}\diff{t}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Fresnelsinint@{z} = \int_{0}^{z}\sin@{\tfrac{1}{2}\pi t^{2}}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>FresnelS(z) = int(sin((1)/(2)*Pi*(t)^(2)), t = 0..z)</syntaxhighlight> || <syntaxhighlight lang=mathematica>FresnelS[z] == Integrate[Sin[Divide[1,2]*Pi*(t)^(2)], {t, 0, z}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
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| [https://dlmf.nist.gov/7.2#Ex3 7.2#Ex3] || [[Item:Q2330|<math>\lim_{x\to\infty}\Fresnelcosint@{x} = \tfrac{1}{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\lim_{x\to\infty}\Fresnelcosint@{x} = \tfrac{1}{2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>limit(FresnelC(x), x = infinity) = (1)/(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Limit[FresnelC[x], x -> Infinity, GenerateConditions->None] == Divide[1,2]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
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| [https://dlmf.nist.gov/7.2#Ex4 7.2#Ex4] || [[Item:Q2331|<math>\lim_{x\to\infty}\Fresnelsinint@{x} = \tfrac{1}{2}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\lim_{x\to\infty}\Fresnelsinint@{x} = \tfrac{1}{2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>limit(FresnelS(x), x = infinity) = (1)/(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Limit[FresnelS[x], x -> Infinity, GenerateConditions->None] == Divide[1,2]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
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| [https://dlmf.nist.gov/7.2.E10 7.2.E10] || [[Item:Q2332|<math>\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}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Fresnelf(z) = ((1)/(2)- FresnelS(z))*cos((1)/(2)*Pi*(z)^(2))-((1)/(2)- FresnelC(z))*sin((1)/(2)*Pi*(z)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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)]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
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| [https://dlmf.nist.gov/7.2.E11 7.2.E11] || [[Item:Q2333|<math>\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}}</math>]]<br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\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}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Fresnelg(z) = ((1)/(2)- FresnelC(z))*cos((1)/(2)*Pi*(z)^(2))+((1)/(2)- FresnelS(z))*sin((1)/(2)*Pi*(z)^(2))</syntaxhighlight> || <syntaxhighlight lang=mathematica>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)]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 7]
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