Error Functions, Dawson’s and Fresnel Integrals - 7.14 Integrals

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
7.14.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}e^{2iat}\erfc@{bt}\diff{t} = {\frac{1}{a\sqrt{\pi}}\DawsonsintF@{\frac{a}{b}}+\frac{i}{2a}\left(1-e^{-(a/b)^{2}}\right)}}
\int_{0}^{\infty}e^{2iat}\erfc@{bt}\diff{t} = {\frac{1}{a\sqrt{\pi}}\DawsonsintF@{\frac{a}{b}}+\frac{i}{2a}\left(1-e^{-(a/b)^{2}}\right)}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\phase@@{b}| < \tfrac{1}{4}\pi} 
int(exp(2*I*a*t)*erfc(b*t), t = 0..infinity) = (1)/(a*sqrt(Pi))*dawson((a)/(b))+(I)/(2*a)*(1 - exp(-(a/b)^(2)))

Integrate[Exp[2*I*a*t]*Erfc[b*t], {t, 0, Infinity}, GenerateConditions->None] == Divide[1,a*Sqrt[Pi]]*DawsonF[Divide[a,b]]+Divide[I,2*a]*(1 - Exp[-(a/b)^(2)])
Failure Aborted Successful [Tested: 18] Successful [Tested: 3]
7.14.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}e^{-at}\erf@{bt}\diff{t} = \frac{1}{a}e^{a^{2}/(4b^{2})}\erfc@{\frac{a}{2b}}}
\int_{0}^{\infty}e^{-at}\erf@{bt}\diff{t} = \frac{1}{a}e^{a^{2}/(4b^{2})}\erfc@{\frac{a}{2b}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{a} > 0, |\phase@@{b}| < \tfrac{1}{4}\pi} 
int(exp(- a*t)*erf(b*t), t = 0..infinity) = (1)/(a)*exp((a)^(2)/(4*(b)^(2)))*erfc((a)/(2*b))

Integrate[Exp[- a*t]*Erf[b*t], {t, 0, Infinity}, GenerateConditions->None] == Divide[1,a]*Exp[(a)^(2)/(4*(b)^(2))]*Erfc[Divide[a,2*b]]
Successful Aborted - Skipped - Because timed out
7.14.E3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}e^{-at}\erf@@{\sqrt{bt}}\diff{t} = \frac{1}{a}\sqrt{\frac{b}{a+b}}}
\int_{0}^{\infty}e^{-at}\erf@@{\sqrt{bt}}\diff{t} = \frac{1}{a}\sqrt{\frac{b}{a+b}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{a} > 0, \realpart@@{b} > 0} 
int(exp(- a*t)*erf(sqrt(b*t)), t = 0..infinity) = (1)/(a)*sqrt((b)/(a + b))

Integrate[Exp[- a*t]*Erf[Sqrt[b*t]], {t, 0, Infinity}, GenerateConditions->None] == Divide[1,a]*Sqrt[Divide[b,a + b]]
Failure Aborted Successful [Tested: 9] Skipped - Because timed out
7.14.E4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}e^{(a-b)t}\erfc@{\sqrt{at}+\sqrt{\frac{c}{t}}}\diff{t} = \frac{e^{-2(\sqrt{ac}+\sqrt{bc})}}{\sqrt{b}(\sqrt{a}+\sqrt{b})}}
\int_{0}^{\infty}e^{(a-b)t}\erfc@{\sqrt{at}+\sqrt{\frac{c}{t}}}\diff{t} = \frac{e^{-2(\sqrt{ac}+\sqrt{bc})}}{\sqrt{b}(\sqrt{a}+\sqrt{b})}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\phase@@{a}| < \frac{1}{2}\pi, \realpart@@{b} > 0, \realpart@@{c} \geq 0} 
int(exp((a - b)*t)*erfc(sqrt(a*t)+sqrt((c)/(t))), t = 0..infinity) = (exp(- 2*(sqrt(a*c)+sqrt(b*c))))/(sqrt(b)*(sqrt(a)+sqrt(b)))

Integrate[Exp[(a - b)*t]*Erfc[Sqrt[a*t]+Sqrt[Divide[c,t]]], {t, 0, Infinity}, GenerateConditions->None] == Divide[Exp[- 2*(Sqrt[a*c]+Sqrt[b*c])],Sqrt[b]*(Sqrt[a]+Sqrt[b])]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
7.14.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}e^{-at}\Fresnelcosint@{t}\diff{t} = \frac{1}{a}\auxFresnelf@{\frac{a}{\pi}}}
\int_{0}^{\infty}e^{-at}\Fresnelcosint@{t}\diff{t} = \frac{1}{a}\auxFresnelf@{\frac{a}{\pi}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{a} > 0} 
int(exp(- a*t)*FresnelC(t), t = 0..infinity) = (1)/(a)*Fresnelf((a)/(Pi))

Integrate[Exp[- a*t]*FresnelC[t], {t, 0, Infinity}, GenerateConditions->None] == Divide[1,a]*FresnelF[Divide[a,Pi]]
Failure Successful Successful [Tested: 3] Successful [Tested: 3]
7.14.E6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}e^{-at}\Fresnelsinint@{t}\diff{t} = \frac{1}{a}\auxFresnelg@{\frac{a}{\pi}}}
\int_{0}^{\infty}e^{-at}\Fresnelsinint@{t}\diff{t} = \frac{1}{a}\auxFresnelg@{\frac{a}{\pi}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{a} > 0} 
int(exp(- a*t)*FresnelS(t), t = 0..infinity) = (1)/(a)*Fresnelg((a)/(Pi))

Integrate[Exp[- a*t]*FresnelS[t], {t, 0, Infinity}, GenerateConditions->None] == Divide[1,a]*FresnelG[Divide[a,Pi]]
Failure Successful Successful [Tested: 3] Successful [Tested: 3]
7.14.E7 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}e^{-at}\Fresnelcosint@{\sqrt{\frac{2t}{\pi}}}\diff{t} = \frac{(\sqrt{a^{2}+1}+a)^{\frac{1}{2}}}{2a\sqrt{a^{2}+1}}}
\int_{0}^{\infty}e^{-at}\Fresnelcosint@{\sqrt{\frac{2t}{\pi}}}\diff{t} = \frac{(\sqrt{a^{2}+1}+a)^{\frac{1}{2}}}{2a\sqrt{a^{2}+1}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{a} > 0} 
int(exp(- a*t)*FresnelC(sqrt((2*t)/(Pi))), t = 0..infinity) = ((sqrt((a)^(2)+ 1)+ a)^((1)/(2)))/(2*a*sqrt((a)^(2)+ 1))

Integrate[Exp[- a*t]*FresnelC[Sqrt[Divide[2*t,Pi]]], {t, 0, Infinity}, GenerateConditions->None] == Divide[(Sqrt[(a)^(2)+ 1]+ a)^(Divide[1,2]),2*a*Sqrt[(a)^(2)+ 1]]
Successful Failure - Successful [Tested: 3]
7.14.E8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}e^{-at}\Fresnelsinint@{\sqrt{\frac{2t}{\pi}}}\diff{t} = \frac{(\sqrt{a^{2}+1}-a)^{\frac{1}{2}}}{2a\sqrt{a^{2}+1}}}
\int_{0}^{\infty}e^{-at}\Fresnelsinint@{\sqrt{\frac{2t}{\pi}}}\diff{t} = \frac{(\sqrt{a^{2}+1}-a)^{\frac{1}{2}}}{2a\sqrt{a^{2}+1}}		 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{a} > 0} 
int(exp(- a*t)*FresnelS(sqrt((2*t)/(Pi))), t = 0..infinity) = ((sqrt((a)^(2)+ 1)- a)^((1)/(2)))/(2*a*sqrt((a)^(2)+ 1))

Integrate[Exp[- a*t]*FresnelS[Sqrt[Divide[2*t,Pi]]], {t, 0, Infinity}, GenerateConditions->None] == Divide[(Sqrt[(a)^(2)+ 1]- a)^(Divide[1,2]),2*a*Sqrt[(a)^(2)+ 1]]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
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