Results of Error Functions, Dawson’s and Fresnel Integrals
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| DLMF | Formula | 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)=(2)/(sqrt(Pi))*int(exp(- (t)^(2)), t = 0..z) |
Erf[z]=Divide[2,Sqrt[Pi]]*Integrate[Exp[- (t)^(2)], {t, 0, z}] |
Successful | Successful | - | - |
| 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)=(2)/(sqrt(Pi))*int(exp(- (t)^(2)), t = z..infinity) |
Erfc[z]=Divide[2,Sqrt[Pi]]*Integrate[Exp[- (t)^(2)], {t, z, Infinity}] |
Successful | Successful | - | - |
| 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}} | (2)/(sqrt(Pi))*int(exp(- (t)^(2)), t = z..infinity)= 1 - erf(z) |
Divide[2,Sqrt[Pi]]*Integrate[Exp[- (t)^(2)], {t, z, Infinity}]= 1 - Erf[z] |
Successful | Successful | - | - |
| 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}} | 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}])= Exp[- (z)^(2)]*Erfc[- I*z] |
Successful | Successful | - | - |
| 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} | limit(erf(z), z = infinity)= 1 |
Limit[Erf[z], z -> Infinity]= 1 |
Successful | Successful | - | - |
| 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} | limit(erfc(z), z = infinity)= 0 |
Limit[Erfc[z], z -> Infinity]= 0 |
Successful | Successful | - | - |
| 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}} | dawson(z)= exp(- (z)^(2))*int(exp((t)^(2)), t = 0..z) |
DawsonF[z]= Exp[- (z)^(2)]*Integrate[Exp[(t)^(2)], {t, 0, z}] |
Successful | Successful | - | - |
| 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}} | 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}] |
Successful | Successful | - | - |
| 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}} | 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}] |
Successful | Successful | - | - |
| 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}} | limit(FresnelC(x), x = infinity)=(1)/(2) |
Limit[FresnelC[x], x -> Infinity]=Divide[1,2] |
Successful | Successful | - | - |
| 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}} | limit(FresnelS(x), x = infinity)=(1)/(2) |
Limit[FresnelS[x], x -> Infinity]=Divide[1,2] |
Successful | Successful | - | - |
| 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}}} | 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 | - | - |
| 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}}} | 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 | - | - |
| 7.4.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \erf@{-z} = -\erf@{z}} | erf(- z)= - erf(z) |
Erf[- z]= - Erf[z] |
Successful | Successful | - | - |
| 7.4.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \erfc@{-z} = 2-\erfc@{z}} | erfc(- z)= 2 - erfc(z) |
Erfc[- z]= 2 - Erfc[z] |
Successful | Successful | - | - |
| 7.4.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \DawsonsintF@{-z} = -\DawsonsintF@{z}} | dawson(- z)= - dawson(z) |
DawsonF[- z]= - DawsonF[z] |
Successful | Successful | - | - |
| 7.4#Ex1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Fresnelcosint@{-z} = -\Fresnelcosint@{z}} | FresnelC(- z)= - FresnelC(z) |
FresnelC[- z]= - FresnelC[z] |
Successful | Successful | - | - |
| 7.4#Ex2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Fresnelsinint@{-z} = -\Fresnelsinint@{z}} | FresnelS(- z)= - FresnelS(z) |
FresnelS[- z]= - FresnelS[z] |
Successful | Successful | - | - |
| 7.4#Ex3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Fresnelcosint@{iz} = i\Fresnelcosint@{z}} | FresnelC(I*z)= I*FresnelC(z) |
FresnelC[I*z]= I*FresnelC[z] |
Successful | Successful | - | - |
| 7.4#Ex4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Fresnelsinint@{iz} = -i\Fresnelsinint@{z}} | FresnelS(I*z)= - I*FresnelS(z) |
FresnelS[I*z]= - I*FresnelS[z] |
Successful | Successful | - | - |
| 7.4#Ex5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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 | Successful |
| 7.4#Ex6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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 | Successful |
| 7.4#Ex7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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] |
Failure | Successful | Successful | - |
| 7.4#Ex8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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] |
Failure | Failure | Successful | Successful |
| 7.5.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Fresnelcosint@{z} = \tfrac{1}{2}+\auxFresnelf@{z}\sin@{\tfrac{1}{2}\pi z^{2}}-\auxFresnelg@{z}\cos@{\tfrac{1}{2}\pi z^{2}}} | FresnelC(z)=(1)/(2)+ Fresnelf(z)*sin((1)/(2)*Pi*(z)^(2))- Fresnelg(z)*cos((1)/(2)*Pi*(z)^(2)) |
FresnelC[z]=Divide[1,2]+ FresnelF[z]*Sin[Divide[1,2]*Pi*(z)^(2)]- FresnelG[z]*Cos[Divide[1,2]*Pi*(z)^(2)] |
Successful | Failure | - | Successful |
| 7.5.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Fresnelsinint@{z} = \tfrac{1}{2}-\auxFresnelf@{z}\cos@{\tfrac{1}{2}\pi z^{2}}-\auxFresnelg@{z}\sin@{\tfrac{1}{2}\pi z^{2}}} | FresnelS(z)=(1)/(2)- Fresnelf(z)*cos((1)/(2)*Pi*(z)^(2))- Fresnelg(z)*sin((1)/(2)*Pi*(z)^(2)) |
FresnelS[z]=Divide[1,2]- FresnelF[z]*Cos[Divide[1,2]*Pi*(z)^(2)]- FresnelG[z]*Sin[Divide[1,2]*Pi*(z)^(2)] |
Successful | Failure | - | Successful |
| 7.5.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle e^{+\frac{1}{2}\pi iz^{2}}(\auxFresnelg@{z}+ i\auxFresnelf@{z}) = \tfrac{1}{2}(1+ i)-(\Fresnelcosint@{z}+ i\Fresnelsinint@{z})} | exp(+(1)/(2)*Pi*I*(z)^(2))*(Fresnelg(z)+ I*Fresnelf(z))=(1)/(2)*(1 + I)-(FresnelC(z)+ I*FresnelS(z)) |
Exp[+Divide[1,2]*Pi*I*(z)^(2)]*(FresnelG[z]+ I*FresnelF[z])=Divide[1,2]*(1 + I)-(FresnelC[z]+ I*FresnelS[z]) |
Failure | Failure | Fail .149314e-2-.173022e-2*I <- {z = 2^(1/2)-I*2^(1/2)}-.119473e-2+.149314e-2*I <- {z = -2^(1/2)+I*2^(1/2)} |
Successful |
| 7.5.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle e^{-\frac{1}{2}\pi iz^{2}}(\auxFresnelg@{z}- i\auxFresnelf@{z}) = \tfrac{1}{2}(1- i)-(\Fresnelcosint@{z}- i\Fresnelsinint@{z})} | exp(-(1)/(2)*Pi*I*(z)^(2))*(Fresnelg(z)- I*Fresnelf(z))=(1)/(2)*(1 - I)-(FresnelC(z)- I*FresnelS(z)) |
Exp[-Divide[1,2]*Pi*I*(z)^(2)]*(FresnelG[z]- I*FresnelF[z])=Divide[1,2]*(1 - I)-(FresnelC[z]- I*FresnelS[z]) |
Failure | Failure | Fail .149314e-2+.173022e-2*I <- {z = 2^(1/2)+I*2^(1/2)}-.119473e-2-.149314e-2*I <- {z = -2^(1/2)-I*2^(1/2)} |
Successful |
| 7.5.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Fresnelcosint@{z}+ i\Fresnelsinint@{z} = \tfrac{1}{2}(1+ i)\erf@@{\zeta}} | FresnelC(z)+ I*FresnelS(z)=(1)/(2)*(1 + I)* erf(zeta) |
FresnelC[z]+ I*FresnelS[z]=Divide[1,2]*(1 + I)* Erf[\[zeta]] |
Failure | Failure | Fail -.1423151062+.1316106532*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)+I*2^(1/2)}.1316106532-.1423151062*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)-I*2^(1/2)}1.141922366+.8679966068*I <- {z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)-I*2^(1/2)}.8679966068+1.141922366*I <- {z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)+I*2^(1/2)}... skip entries to safe data |
Error |
| 7.5.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Fresnelcosint@{z}- i\Fresnelsinint@{z} = \tfrac{1}{2}(1- i)\erf@@{\zeta}} | FresnelC(z)- I*FresnelS(z)=(1)/(2)*(1 - I)* erf(zeta) |
FresnelC[z]- I*FresnelS[z]=Divide[1,2]*(1 - I)* Erf[\[zeta]] |
Failure | Failure | Fail 66.79933367+67.80964539*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)+I*2^(1/2)}66.52540791+67.53571963*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)-I*2^(1/2)}67.53571963+66.52540791*I <- {z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)-I*2^(1/2)}67.80964539+66.79933367*I <- {z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)+I*2^(1/2)}... skip entries to safe data |
Error |
| 7.5.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \auxFresnelg@{z}+ i\auxFresnelf@{z} = \tfrac{1}{2}(1+ i)e^{\zeta^{2}}\erfc@@{\zeta}} | Fresnelg(z)+ I*Fresnelf(z)=(1)/(2)*(1 + I)* exp((zeta)^(2))*erfc(zeta) |
FresnelG[z]+ I*FresnelF[z]=Divide[1,2]*(1 + I)* Exp[(\[zeta])^(2)]*Erfc[\[zeta]] |
Failure | Failure | Fail -.874918896e-1+.8375300635e-1*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)+I*2^(1/2)}.8375400635e-1-.874928896e-1*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)-I*2^(1/2)}.1946430180+1.537001110*I <- {z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)-I*2^(1/2)}1.537002110+.1946420180*I <- {z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)+I*2^(1/2)}... skip entries to safe data |
Error |
| 7.5.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \auxFresnelg@{z}- i\auxFresnelf@{z} = \tfrac{1}{2}(1- i)e^{\zeta^{2}}\erfc@@{\zeta}} | Fresnelg(z)- I*Fresnelf(z)=(1)/(2)*(1 - I)* exp((zeta)^(2))*erfc(zeta) |
FresnelG[z]- I*FresnelF[z]=Divide[1,2]*(1 - I)* Exp[(\[zeta])^(2)]*Erfc[\[zeta]] |
Failure | Failure | Fail -.1458959936+.662848896e-1*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)+I*2^(1/2)}-.3171418896-.1049610064*I <- {z = 2^(1/2)+I*2^(1/2), zeta = 2^(1/2)-I*2^(1/2)}1.307352110-.2158500180*I <- {z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)-I*2^(1/2)}-.3500698200e-1-1.558209110*I <- {z = 2^(1/2)+I*2^(1/2), zeta = -2^(1/2)+I*2^(1/2)}... skip entries to safe data |
Error |
| 7.6.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \erf@@{z} = \frac{2}{\sqrt{\pi}}\sum_{n=0}^{\infty}\frac{(-1)^{n}z^{2n+1}}{n!(2n+1)}} | erf(z)=(2)/(sqrt(Pi))*sum(((- 1)^(n)* (z)^(2*n + 1))/(factorial(n)*(2*n + 1)), n = 0..infinity) |
Erf[z]=Divide[2,Sqrt[Pi]]*Sum[Divide[(- 1)^(n)* (z)^(2*n + 1),(n)!*(2*n + 1)], {n, 0, Infinity}] |
Successful | Successful | - | - |
| 7.6.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Fresnelcosint@{z} = \sum_{n=0}^{\infty}\frac{(-1)^{n}(\frac{1}{2}\pi)^{2n}}{(2n)!(4n+1)}z^{4n+1}} | FresnelC(z)= sum(((- 1)^(n)*((1)/(2)*Pi)^(2*n))/(factorial(2*n)*(4*n + 1))*(z)^(4*n + 1), n = 0..infinity) |
FresnelC[z]= Sum[Divide[(- 1)^(n)*(Divide[1,2]*Pi)^(2*n),(2*n)!*(4*n + 1)]*(z)^(4*n + 1), {n, 0, Infinity}] |
Successful | Successful | - | - |
| 7.6.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Fresnelsinint@{z} = \sum_{n=0}^{\infty}\frac{(-1)^{n}(\frac{1}{2}\pi)^{2n+1}}{(2n+1)!(4n+3)}z^{4n+3}} | FresnelS(z)= sum(((- 1)^(n)*((1)/(2)*Pi)^(2*n + 1))/(factorial(2*n + 1)*(4*n + 3))*(z)^(4*n + 3), n = 0..infinity) |
FresnelS[z]= Sum[Divide[(- 1)^(n)*(Divide[1,2]*Pi)^(2*n + 1),(2*n + 1)!*(4*n + 3)]*(z)^(4*n + 3), {n, 0, Infinity}] |
Successful | Successful | - | - |
| 7.6.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \erf@@{z} = \frac{2z}{\sqrt{\pi}}\sum_{n=0}^{\infty}(-1)^{n}\left(\modsphBesseli{1}{2n}@{z^{2}}-\modsphBesseli{1}{2n+1}@{z^{2}}\right)} | Error |
\|Sqrt[1/2 Pi /$2] BesselI[-2*n - 1/2, 2*n]*(z)^(2)- Sqrt[1/2 Pi /$2] BesselI[(-1)^(1-1)*2*n + 1 + 1/2, 2*n + 1]\|\|Sqrt[1/2 Pi /$2] BesselI[-2*n + 1 - 1/2, 2*n + 1]*(z)^(2)), {n, 0, Infinity}] | Error | Error | - | - |
| 7.6.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \erf@{az} = \frac{2z}{\sqrt{\pi}}e^{(\frac{1}{2}-a^{2})z^{2}}\sum_{n=0}^{\infty}\ChebyshevpolyT{2n+1}@{a}\modsphBesseli{1}{n}@{\tfrac{1}{2}z^{2}}} | Error |
\|Sqrt[1/2 Pi /$2] BesselI[-n - 1/2, n]*Divide[1,2]*(z)^(2), {n, 0, Infinity}] | Error | Error | - | - |
| 7.6.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Fresnelcosint@{z} = z\sum_{n=0}^{\infty}\sphBesselJ{2n}@{\tfrac{1}{2}\pi z^{2}}} | Error |
FresnelC[z]= z*Sum[SphericalBesselJ[2*n, Divide[1,2]*Pi*(z)^(2)], {n, 0, Infinity}] |
Error | Failure | - | Skip |
| 7.6.E11 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Fresnelsinint@{z} = z\sum_{n=0}^{\infty}\sphBesselJ{2n+1}@{\tfrac{1}{2}\pi z^{2}}} | Error |
FresnelS[z]= z*Sum[SphericalBesselJ[2*n + 1, Divide[1,2]*Pi*(z)^(2)], {n, 0, Infinity}] |
Error | Failure | - | Skip |
| 7.7.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \erfc@@{z} = \frac{2}{\pi}e^{-z^{2}}\int_{0}^{\infty}\frac{e^{-z^{2}t^{2}}}{t^{2}+1}\diff{t}} | erfc(z)=(2)/(Pi)*exp(- (z)^(2))*int((exp(- (z)^(2)* (t)^(2)))/((t)^(2)+ 1), t = 0..infinity) |
Erfc[z]=Divide[2,Pi]*Exp[- (z)^(2)]*Integrate[Divide[Exp[- (z)^(2)* (t)^(2)],(t)^(2)+ 1], {t, 0, Infinity}] |
Successful | Failure | - | Error |
| 7.7.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{\pi i}\int_{-\infty}^{\infty}\frac{e^{-t^{2}}\diff{t}}{t-z} = \frac{2z}{\pi i}\int_{0}^{\infty}\frac{e^{-t^{2}}\diff{t}}{t^{2}-z^{2}}} | (1)/(Pi*I)*int((exp(- (t)^(2)))/(t - z), t = - infinity..infinity)=(2*z)/(Pi*I)*int((exp(- (t)^(2)))/((t)^(2)- (z)^(2)), t = 0..infinity) |
Divide[1,Pi*I]*Integrate[Divide[Exp[- (t)^(2)],t - z], {t, - Infinity, Infinity}]=Divide[2*z,Pi*I]*Integrate[Divide[Exp[- (t)^(2)],(t)^(2)- (z)^(2)], {t, 0, Infinity}] |
Failure | Failure | Skip | Successful |
| 7.7.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}e^{-at^{2}+2izt}\diff{t} = \frac{1}{2}\sqrt{\frac{\pi}{a}}e^{-z^{2}/a}+\frac{i}{\sqrt{a}}\DawsonsintF@{\frac{z}{\sqrt{a}}}} | int(exp(- a*(t)^(2)+ 2*I*z*t), t = 0..infinity)=(1)/(2)*sqrt((Pi)/(a))*exp(- (z)^(2)/ a)+(I)/(sqrt(a))*dawson((z)/(sqrt(a))) |
Integrate[Exp[- a*(t)^(2)+ 2*I*z*t], {t, 0, Infinity}]=Divide[1,2]*Sqrt[Divide[Pi,a]]*Exp[- (z)^(2)/ a]+Divide[I,Sqrt[a]]*DawsonF[Divide[z,Sqrt[a]]] |
Failure | Successful | Skip | - |
| 7.7.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}\frac{e^{-at}}{\sqrt{t+z^{2}}}\diff{t} = \sqrt{\frac{\pi}{a}}e^{az^{2}}\erfc@{\sqrt{a}z}} | int((exp(- a*t))/(sqrt(t + (z)^(2))), t = 0..infinity)=sqrt((Pi)/(a))*exp(a*(z)^(2))*erfc(sqrt(a)*z) |
Integrate[Divide[Exp[- a*t],Sqrt[t + (z)^(2)]], {t, 0, Infinity}]=Sqrt[Divide[Pi,a]]*Exp[a*(z)^(2)]*Erfc[Sqrt[a]*z] |
Successful | Failure | - | Successful |
| 7.7.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{x}^{\infty}e^{-(at^{2}+2bt+c)}\diff{t} = \frac{1}{2}\sqrt{\frac{\pi}{a}}e^{(b^{2}-ac)/a}\erfc@{\sqrt{a}x+\frac{b}{\sqrt{a}}}} | int(exp(-(a*(t)^(2)+ 2*b*t + c)), t = x..infinity)=(1)/(2)*sqrt((Pi)/(a))*exp(((b)^(2)- a*c)/ a)*erfc(sqrt(a)*x +(b)/(sqrt(a))) |
Integrate[Exp[-(a*(t)^(2)+ 2*b*t + c)], {t, x, Infinity}]=Divide[1,2]*Sqrt[Divide[Pi,a]]*Exp[((b)^(2)- a*c)/ a]*Erfc[Sqrt[a]*x +Divide[b,Sqrt[a]]] |
Failure | Failure | Skip | Successful |
| 7.7.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{x}^{\infty}e^{-a^{2}t^{2}-(b^{2}/t^{2})}\diff{t} = \frac{\sqrt{\pi}}{4a}\left(e^{2ab}\erfc@{ax+(b/x)}+e^{-2ab}\erfc@{ax-(b/x)}\right)} | int(exp(- (a)^(2)* (t)^(2)-((b)^(2)/ (t)^(2))), t = x..infinity)=(sqrt(Pi))/(4*a)*(exp(2*a*b)*erfc(a*x +(b/ x))+ exp(- 2*a*b)*erfc(a*x -(b/ x))) |
Integrate[Exp[- (a)^(2)* (t)^(2)-((b)^(2)/ (t)^(2))], {t, x, Infinity}]=Divide[Sqrt[Pi],4*a]*(Exp[2*a*b]*Erfc[a*x +(b/ x)]+ Exp[- 2*a*b]*Erfc[a*x -(b/ x)]) |
Failure | Failure | Skip | Error |
| 7.7.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}e^{-a^{2}t^{2}-(b^{2}/t^{2})}\diff{t} = \frac{\sqrt{\pi}}{2a}e^{-2ab}} | int(exp(- (a)^(2)* (t)^(2)-((b)^(2)/ (t)^(2))), t = 0..infinity)=(sqrt(Pi))/(2*a)*exp(- 2*a*b) |
Integrate[Exp[- (a)^(2)* (t)^(2)-((b)^(2)/ (t)^(2))], {t, 0, Infinity}]=Divide[Sqrt[Pi],2*a]*Exp[- 2*a*b] |
Successful | Failure | - | Successful |
| 7.7.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{x}\erf@@{t}\diff{t} = x\erf@@{x}+\frac{1}{\sqrt{\pi}}\left(e^{-x^{2}}-1\right)} | int(erf(t), t = 0..x)= x*erf(x)+(1)/(sqrt(Pi))*(exp(- (x)^(2))- 1) |
Integrate[Erf[t], {t, 0, x}]= x*Erf[x]+Divide[1,Sqrt[Pi]]*(Exp[- (x)^(2)]- 1) |
Successful | Successful | - | - |
| 7.7.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \auxFresnelf@{z} = \frac{1}{\pi\sqrt{2}}\int_{0}^{\infty}\frac{e^{-\pi z^{2}t/2}}{\sqrt{t}(t^{2}+1)}\diff{t}} | Fresnelf(z)=(1)/(Pi*sqrt(2))*int((exp(- Pi*(z)^(2)* t/ 2))/(sqrt(t)*((t)^(2)+ 1)), t = 0..infinity) |
FresnelF[z]=Divide[1,Pi*Sqrt[2]]*Integrate[Divide[Exp[- Pi*(z)^(2)* t/ 2],Sqrt[t]*((t)^(2)+ 1)], {t, 0, Infinity}] |
Failure | Failure | Skip | Error |
| 7.7.E11 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \auxFresnelg@{z} = \frac{1}{\pi\sqrt{2}}\int_{0}^{\infty}\frac{\sqrt{t}e^{-\pi z^{2}t/2}}{t^{2}+1}\diff{t}} | Fresnelg(z)=(1)/(Pi*sqrt(2))*int((sqrt(t)*exp(- Pi*(z)^(2)* t/ 2))/((t)^(2)+ 1), t = 0..infinity) |
FresnelG[z]=Divide[1,Pi*Sqrt[2]]*Integrate[Divide[Sqrt[t]*Exp[- Pi*(z)^(2)* t/ 2],(t)^(2)+ 1], {t, 0, Infinity}] |
Failure | Failure | Skip | Error |
| 7.7.E12 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \auxFresnelg@{z}+i\auxFresnelf@{z} = e^{-\pi iz^{2}/2}\int_{z}^{\infty}e^{\pi it^{2}/2}\diff{t}} | Fresnelg(z)+ I*Fresnelf(z)= exp(- Pi*I*(z)^(2)/ 2)*int(exp(Pi*I*(t)^(2)/ 2), t = z..infinity) |
FresnelG[z]+ I*FresnelF[z]= Exp[- Pi*I*(z)^(2)/ 2]*Integrate[Exp[Pi*I*(t)^(2)/ 2], {t, z, Infinity}] |
Failure | Failure | Skip | Fail
Complex[-0.12449815517713354, 0.12449815517716199] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[0.12449815517710515, -0.12449815517713354] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} |
| 7.7.E13 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \auxFresnelf@{z} = \frac{(2\pi)^{-3/2}}{2\pi i}\int_{c-i\infty}^{c+i\infty}\zeta^{-s}\EulerGamma@{s}\EulerGamma@{s+\tfrac{1}{2}}\*\EulerGamma@{s+\tfrac{3}{4}}\EulerGamma@{\tfrac{1}{4}-s}\diff{s}} | Fresnelf(z)=((2*Pi)^(- 3/ 2))/(2*Pi*I)*int((zeta)^(- s)* GAMMA(s)*GAMMA(s +(1)/(2))* GAMMA(s +(3)/(4))*GAMMA((1)/(4)- s), s = c - I*infinity..c + I*infinity) |
FresnelF[z]=Divide[(2*Pi)^(- 3/ 2),2*Pi*I]*Integrate[(\[zeta])^(- s)* Gamma[s]*Gamma[s +Divide[1,2]]* Gamma[s +Divide[3,4]]*Gamma[Divide[1,4]- s], {s, c - I*Infinity, c + I*Infinity}] |
Failure | Failure | Skip | Error |
| 7.7.E14 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \auxFresnelg@{z} = \frac{(2\pi)^{-3/2}}{2\pi i}\int_{c-i\infty}^{c+i\infty}\zeta^{-s}\EulerGamma@{s}\EulerGamma@{s+\tfrac{1}{2}}\*\EulerGamma@{s+\tfrac{1}{4}}\EulerGamma@{\tfrac{3}{4}-s}\diff{s}} | Fresnelg(z)=((2*Pi)^(- 3/ 2))/(2*Pi*I)*int((zeta)^(- s)* GAMMA(s)*GAMMA(s +(1)/(2))* GAMMA(s +(1)/(4))*GAMMA((3)/(4)- s), s = c - I*infinity..c + I*infinity) |
FresnelG[z]=Divide[(2*Pi)^(- 3/ 2),2*Pi*I]*Integrate[(\[zeta])^(- s)* Gamma[s]*Gamma[s +Divide[1,2]]* Gamma[s +Divide[1,4]]*Gamma[Divide[3,4]- s], {s, c - I*Infinity, c + I*Infinity}] |
Failure | Failure | Skip | Error |
| 7.7.E15 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}e^{-at}\cos@{t^{2}}\diff{t} = \sqrt{\frac{\pi}{2}}\auxFresnelf@{\frac{a}{\sqrt{2\pi}}}} | int(exp(- a*t)*cos((t)^(2)), t = 0..infinity)=sqrt((Pi)/(2))*Fresnelf((a)/(sqrt(2*Pi))) |
Integrate[Exp[- a*t]*Cos[(t)^(2)], {t, 0, Infinity}]=Sqrt[Divide[Pi,2]]*FresnelF[Divide[a,Sqrt[2*Pi]]] |
Successful | Failure | - | Error |
| 7.7.E16 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{\infty}e^{-at}\sin@{t^{2}}\diff{t} = \sqrt{\frac{\pi}{2}}\auxFresnelg@{\frac{a}{\sqrt{2\pi}}}} | int(exp(- a*t)*sin((t)^(2)), t = 0..infinity)=sqrt((Pi)/(2))*Fresnelg((a)/(sqrt(2*Pi))) |
Integrate[Exp[- a*t]*Sin[(t)^(2)], {t, 0, Infinity}]=Sqrt[Divide[Pi,2]]*FresnelG[Divide[a,Sqrt[2*Pi]]] |
Successful | Failure | - | Error |
| 7.8.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{\int_{x}^{\infty}e^{-t^{2}}\diff{t}}{e^{-x^{2}}} = e^{x^{2}}\int_{x}^{\infty}e^{-t^{2}}\diff{t}} | (int(exp(- (t)^(2)), t = x..infinity))/(exp(- (x)^(2)))= exp((x)^(2))*int(exp(- (t)^(2)), t = x..infinity) |
Divide[Integrate[Exp[- (t)^(2)], {t, x, Infinity}],Exp[- (x)^(2)]]= Exp[(x)^(2)]*Integrate[Exp[- (t)^(2)], {t, x, Infinity}] |
Successful | Successful | - | - |
| 7.8.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{x^{2}}{2x^{2}+1} <= \frac{x^{2}(2x^{2}+5)}{4x^{4}+12x^{2}+3}} | ((x)^(2))/(2*(x)^(2)+ 1)< =((x)^(2)*(2*(x)^(2)+ 5))/(4*(x)^(4)+ 12*(x)^(2)+ 3) |
Divide[(x)^(2),2*(x)^(2)+ 1]< =Divide[(x)^(2)*(2*(x)^(2)+ 5),4*(x)^(4)+ 12*(x)^(2)+ 3] |
Failure | Failure | Successful | Successful |
| 7.8.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{2x^{4}+9x^{2}+4}{4x^{4}+20x^{2}+15} < \frac{x^{2}+1}{2x^{2}+3}} | (2*(x)^(4)+ 9*(x)^(2)+ 4)/(4*(x)^(4)+ 20*(x)^(2)+ 15)<((x)^(2)+ 1)/(2*(x)^(2)+ 3) |
Divide[2*(x)^(4)+ 9*(x)^(2)+ 4,4*(x)^(4)+ 20*(x)^(2)+ 15]<Divide[(x)^(2)+ 1,2*(x)^(2)+ 3] |
Failure | Failure | Skip | Successful |
| 7.8.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{x}e^{at^{2}}\diff{t} < \frac{1}{3ax}\left(2e^{ax^{2}}+ax^{2}-2\right)} | int(exp(a*(t)^(2)), t = 0..x)<(1)/(3*a*x)*(2*exp(a*(x)^(2))+ a*(x)^(2)- 2) |
Integrate[Exp[a*(t)^(2)], {t, 0, x}]<Divide[1,3*a*x]*(2*Exp[a*(x)^(2)]+ a*(x)^(2)- 2) |
Error | Failure | - | Successful |
| 7.8.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{0}^{x}e^{t^{2}}\diff{t} < \frac{e^{x^{2}}-1}{x}} | int(exp((t)^(2)), t = 0..x)<(exp((x)^(2))- 1)/(x) |
Integrate[Exp[(t)^(2)], {t, 0, x}]<Divide[Exp[(x)^(2)]- 1,x] |
Failure | Failure | Skip | Successful |
| 7.8.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \erf@@{x} < \sqrt{1-\expe^{-4x^{2}/\cpi}}} | erf(x)<sqrt(1 - exp(- 4*(x)^(2)/ Pi)) |
Erf[x]<Sqrt[1 - Exp[- 4*(x)^(2)/ Pi]] |
Failure | Failure | Skip | Successful |
| 7.10.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \deriv[n+1]{\erf@@{z}}{z} = (-1)^{n}\frac{2}{\sqrt{\pi}}\HermitepolyH{n}@{z}e^{-z^{2}}} | diff(erf(z), [z$(n + 1)])=(- 1)^(n)*(2)/(sqrt(Pi))*HermiteH(n, z)*exp(- (z)^(2)) |
D[Erf[z], {z, n + 1}]=(- 1)^(n)*Divide[2,Sqrt[Pi]]*HermiteH[n, z]*Exp[- (z)^(2)] |
Failure | Failure | Skip | Successful |
| 7.10#Ex1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \deriv{\auxFresnelf@{z}}{z} = -\pi z\auxFresnelg@{z}} | diff(Fresnelf(z), z)= - Pi*z*Fresnelg(z) |
D[FresnelF[z], z]= - Pi*z*FresnelG[z] |
Successful | Successful | - | - |
| 7.10#Ex2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \deriv{\auxFresnelg@{z}}{z} = \pi z\auxFresnelf@{z}-1} | diff(Fresnelg(z), z)= Pi*z*Fresnelf(z)- 1 |
D[FresnelG[z], z]= Pi*z*FresnelF[z]- 1 |
Successful | Successful | - | - |
| 7.11.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \erf@@{z} = \frac{1}{\sqrt{\pi}}\incgamma@{\tfrac{1}{2}}{z^{2}}} | erf(z)=(1)/(sqrt(Pi))*GAMMA((1)/(2))-GAMMA((1)/(2), (z)^(2)) |
Erf[z]=Divide[1,Sqrt[Pi]]*Gamma[Divide[1,2], 0, (z)^(2)] |
Failure | Failure | Fail -.796532174e-2+.2115950078*I <- {z = 2^(1/2)+I*2^(1/2)}-.796532174e-2-.2115950078*I <- {z = 2^(1/2)-I*2^(1/2)}-2.028588748+.7594465268*I <- {z = -2^(1/2)-I*2^(1/2)}-2.028588748-.7594465268*I <- {z = -2^(1/2)+I*2^(1/2)} |
Fail
Complex[-2.020623424050978, 0.547851518927081] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[-2.020623424050978, -0.547851518927081] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} |
| 7.11.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \erfc@@{z} = \frac{1}{\sqrt{\pi}}\incGamma@{\tfrac{1}{2}}{z^{2}}} | erfc(z)=(1)/(sqrt(Pi))*GAMMA((1)/(2), (z)^(2)) |
Erfc[z]=Divide[1,Sqrt[Pi]]*Gamma[Divide[1,2], (z)^(2)] |
Failure | Failure | Fail 2.020623426-.5478515190*I <- {z = -2^(1/2)-I*2^(1/2)}2.020623426+.5478515190*I <- {z = -2^(1/2)+I*2^(1/2)} |
Fail
Complex[2.0206234240509775, -0.547851518927081] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[2.0206234240509775, 0.547851518927081] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} |
| 7.11.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \erfc@@{z} = \frac{z}{\sqrt{\pi}}\genexpintE{\frac{1}{2}}@{z^{2}}} | erfc(z)=(z)/(sqrt(Pi))*Ei((1)/(2), (z)^(2)) |
Erfc[z]=Divide[z,Sqrt[Pi]]*ExpIntegralE[Divide[1,2], (z)^(2)] |
Failure | Failure | Fail 2.000000000-.1e-9*I <- {z = -2^(1/2)-I*2^(1/2)}2.000000000+.1e-9*I <- {z = -2^(1/2)+I*2^(1/2)} |
Fail
Complex[1.9999999999999996, -1.1102230246251565*^-16] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[1.9999999999999996, 1.1102230246251565*^-16] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} |
| 7.11.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \erf@@{z} = \frac{2z}{\sqrt{\pi}}\KummerconfhyperM@{\tfrac{1}{2}}{\tfrac{3}{2}}{-z^{2}}} | erf(z)=(2*z)/(sqrt(Pi))*KummerM((1)/(2), (3)/(2), - (z)^(2)) |
Erf[z]=Divide[2*z,Sqrt[Pi]]*Hypergeometric1F1[Divide[1,2], Divide[3,2], - (z)^(2)] |
Successful | Successful | - | - |
| 7.11.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{2z}{\sqrt{\pi}}\KummerconfhyperM@{\tfrac{1}{2}}{\tfrac{3}{2}}{-z^{2}} = \frac{2z}{\sqrt{\pi}}e^{-z^{2}}\KummerconfhyperM@{1}{\tfrac{3}{2}}{z^{2}}} | (2*z)/(sqrt(Pi))*KummerM((1)/(2), (3)/(2), - (z)^(2))=(2*z)/(sqrt(Pi))*exp(- (z)^(2))*KummerM(1, (3)/(2), (z)^(2)) |
Divide[2*z,Sqrt[Pi]]*Hypergeometric1F1[Divide[1,2], Divide[3,2], - (z)^(2)]=Divide[2*z,Sqrt[Pi]]*Exp[- (z)^(2)]*Hypergeometric1F1[1, Divide[3,2], (z)^(2)] |
Successful | Successful | - | - |
| 7.11.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \erfc@@{z} = \frac{1}{\sqrt{\pi}}e^{-z^{2}}\KummerconfhyperU@{\tfrac{1}{2}}{\tfrac{1}{2}}{z^{2}}} | erfc(z)=(1)/(sqrt(Pi))*exp(- (z)^(2))*KummerU((1)/(2), (1)/(2), (z)^(2)) |
Erfc[z]=Divide[1,Sqrt[Pi]]*Exp[- (z)^(2)]*HypergeometricU[Divide[1,2], Divide[1,2], (z)^(2)] |
Failure | Failure | Fail 2.020623426-.5478515190*I <- {z = -2^(1/2)-I*2^(1/2)}2.020623426+.5478515190*I <- {z = -2^(1/2)+I*2^(1/2)} |
Fail
Complex[2.0206234240509775, -0.547851518927081] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[2.0206234240509775, 0.547851518927081] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} |
| 7.11.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{\sqrt{\pi}}e^{-z^{2}}\KummerconfhyperU@{\tfrac{1}{2}}{\tfrac{1}{2}}{z^{2}} = \frac{z}{\sqrt{\pi}}e^{-z^{2}}\KummerconfhyperU@{1}{\tfrac{3}{2}}{z^{2}}} | (1)/(sqrt(Pi))*exp(- (z)^(2))*KummerU((1)/(2), (1)/(2), (z)^(2))=(z)/(sqrt(Pi))*exp(- (z)^(2))*KummerU(1, (3)/(2), (z)^(2)) |
Divide[1,Sqrt[Pi]]*Exp[- (z)^(2)]*HypergeometricU[Divide[1,2], Divide[1,2], (z)^(2)]=Divide[z,Sqrt[Pi]]*Exp[- (z)^(2)]*HypergeometricU[1, Divide[3,2], (z)^(2)] |
Failure | Failure | Fail -.2062342514e-1+.5478515190*I <- {z = -2^(1/2)-I*2^(1/2)}-.2062342514e-1-.5478515190*I <- {z = -2^(1/2)+I*2^(1/2)} |
Fail
Complex[-0.02062342405097809, 0.5478515189270807] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[-0.020623424050978133, -0.5478515189270807] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} |
| 7.11.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Fresnelcosint@{z}+i\Fresnelsinint@{z} = z\KummerconfhyperM@{\tfrac{1}{2}}{\tfrac{3}{2}}{\tfrac{1}{2}\pi iz^{2}}} | FresnelC(z)+ I*FresnelS(z)= z*KummerM((1)/(2), (3)/(2), (1)/(2)*Pi*I*(z)^(2)) |
FresnelC[z]+ I*FresnelS[z]= z*Hypergeometric1F1[Divide[1,2], Divide[3,2], Divide[1,2]*Pi*I*(z)^(2)] |
Failure | Successful | Successful | - |
| 7.11.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle z\KummerconfhyperM@{\tfrac{1}{2}}{\tfrac{3}{2}}{\tfrac{1}{2}\pi iz^{2}} = ze^{\pi iz^{2}/2}\KummerconfhyperM@{1}{\tfrac{3}{2}}{-\tfrac{1}{2}\pi iz^{2}}} | z*KummerM((1)/(2), (3)/(2), (1)/(2)*Pi*I*(z)^(2))= z*exp(Pi*I*(z)^(2)/ 2)*KummerM(1, (3)/(2), -(1)/(2)*Pi*I*(z)^(2)) |
z*Hypergeometric1F1[Divide[1,2], Divide[3,2], Divide[1,2]*Pi*I*(z)^(2)]= z*Exp[Pi*I*(z)^(2)/ 2]*Hypergeometric1F1[1, Divide[3,2], -Divide[1,2]*Pi*I*(z)^(2)] |
Successful | Successful | - | - |
| 7.11.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Fresnelcosint@{z} = z\genhyperF{1}{2}@{\tfrac{1}{4}}{\tfrac{5}{4},\tfrac{1}{2}}{-\tfrac{1}{16}\pi^{2}z^{4}}} | FresnelC(z)= z*hypergeom([(1)/(4)], [(5)/(4),(1)/(2)], -(1)/(16)*(Pi)^(2)* (z)^(4)) |
FresnelC[z]= z*HypergeometricPFQ[{Divide[1,4]}, {Divide[5,4],Divide[1,2]}, -Divide[1,16]*(Pi)^(2)* (z)^(4)] |
Successful | Successful | - | - |
| 7.11.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Fresnelsinint@{z} = \tfrac{1}{6}\pi z^{3}\genhyperF{1}{2}@{\tfrac{3}{4}}{\tfrac{7}{4},\tfrac{3}{2}}{-\tfrac{1}{16}\pi^{2}z^{4}}} | FresnelS(z)=(1)/(6)*Pi*(z)^(3)* hypergeom([(3)/(4)], [(7)/(4),(3)/(2)], -(1)/(16)*(Pi)^(2)* (z)^(4)) |
FresnelS[z]=Divide[1,6]*Pi*(z)^(3)* HypergeometricPFQ[{Divide[3,4]}, {Divide[7,4],Divide[3,2]}, -Divide[1,16]*(Pi)^(2)* (z)^(4)] |
Successful | Successful | - | - |
| 7.13#Ex4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \mu = \ln@{\lambda\sqrt{2\pi}}} | mu = ln(lambda*sqrt(2*Pi)) |
\[Mu]= Log[\[Lambda]*Sqrt[2*Pi]] |
Failure | Failure | Fail -.197872151+.6288153986*I <- {lambda = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2)}-.197872151-2.199611725*I <- {lambda = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)-I*2^(1/2)}-3.026299275-2.199611725*I <- {lambda = 2^(1/2)+I*2^(1/2), mu = -2^(1/2)-I*2^(1/2)}-3.026299275+.6288153986*I <- {lambda = 2^(1/2)+I*2^(1/2), mu = -2^(1/2)+I*2^(1/2)}... skip entries to safe data |
Fail
Complex[-0.1978721513915227, 0.6288153989756469] <- {Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[-0.1978721513915227, -2.199611725770543] <- {Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[μ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[-3.026299276137713, -2.199611725770543] <- {Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[μ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[-3.026299276137713, 0.6288153989756469] <- {Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[μ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 7.13#Ex8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \mu = \ln@{2\lambda\sqrt{2\pi}}} | mu = ln(2*lambda*sqrt(2*Pi)) |
\[Mu]= Log[2*\[Lambda]*Sqrt[2*Pi]] |
Failure | Failure | Fail -.891019332+.6288153986*I <- {lambda = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)+I*2^(1/2)}-.891019332-2.199611725*I <- {lambda = 2^(1/2)+I*2^(1/2), mu = 2^(1/2)-I*2^(1/2)}-3.719446456-2.199611725*I <- {lambda = 2^(1/2)+I*2^(1/2), mu = -2^(1/2)-I*2^(1/2)}-3.719446456+.6288153986*I <- {lambda = 2^(1/2)+I*2^(1/2), mu = -2^(1/2)+I*2^(1/2)}... skip entries to safe data |
Fail
Complex[-0.8910193319514683, 0.6288153989756469] <- {Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[μ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[-0.8910193319514683, -2.199611725770543] <- {Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[μ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[-3.719446456697659, -2.199611725770543] <- {Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[μ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[-3.719446456697659, 0.6288153989756469] <- {Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[μ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 7.13#Ex12 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \alpha = (2/\pi)\ln@{\pi\lambda}} | alpha =(2/ Pi)* ln(Pi*lambda) |
\[Alpha]=(2/ Pi)* Log[Pi*\[Lambda]] |
Failure | Failure | Fail .244184683+.9142135621*I <- {alpha = 2^(1/2)+I*2^(1/2), lambda = 2^(1/2)+I*2^(1/2)}.244184683+1.914213562*I <- {alpha = 2^(1/2)+I*2^(1/2), lambda = 2^(1/2)-I*2^(1/2)}.244184683+2.914213561*I <- {alpha = 2^(1/2)+I*2^(1/2), lambda = -2^(1/2)-I*2^(1/2)}.244184683-.85786437e-1*I <- {alpha = 2^(1/2)+I*2^(1/2), lambda = -2^(1/2)+I*2^(1/2)}... skip entries to safe data |
Fail
Complex[0.24418468271597948, 0.9142135623730951] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[0.24418468271597948, 1.9142135623730951] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[λ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[0.24418468271597948, 2.914213562373095] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[λ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[0.24418468271597948, -0.08578643762690485] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[λ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 7.13#Ex14 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \alpha = (2/\pi)\ln@{\pi\lambda}} | alpha =(2/ Pi)* ln(Pi*lambda) |
\[Alpha]=(2/ Pi)* Log[Pi*\[Lambda]] |
Failure | Failure | Fail .244184683+.9142135621*I <- {alpha = 2^(1/2)+I*2^(1/2), lambda = 2^(1/2)+I*2^(1/2)}.244184683+1.914213562*I <- {alpha = 2^(1/2)+I*2^(1/2), lambda = 2^(1/2)-I*2^(1/2)}.244184683+2.914213561*I <- {alpha = 2^(1/2)+I*2^(1/2), lambda = -2^(1/2)-I*2^(1/2)}.244184683-.85786437e-1*I <- {alpha = 2^(1/2)+I*2^(1/2), lambda = -2^(1/2)+I*2^(1/2)}... skip entries to safe data |
Fail
Complex[0.24418468271597948, 0.9142135623730951] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[λ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[0.24418468271597948, 1.9142135623730951] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[λ, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[0.24418468271597948, 2.914213562373095] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[λ, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[0.24418468271597948, -0.08578643762690485] <- {Rule[α, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[λ, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 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(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}]=Divide[1,a*Sqrt[Pi]]*DawsonF[Divide[a,b]]+Divide[I,2*a]*(1 - Exp[-(a/ b)^(2)]) |
Failure | Failure | Skip | Error |
| 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(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}]=Divide[1,a]*Exp[(a)^(2)/(4*(b)^(2))]*Erfc[Divide[a,2*b]] |
Successful | Failure | - | Error |
| 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(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}]=Divide[Exp[- 2*(Sqrt[a*c]+Sqrt[b*c])],Sqrt[b]*(Sqrt[a]+Sqrt[b])] |
Failure | Failure | Skip | Error |
| 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(exp(- a*t)*FresnelC(t), t = 0..infinity)=(1)/(a)*Fresnelf((a)/(Pi)) |
Integrate[Exp[- a*t]*FresnelC[t], {t, 0, Infinity}]=Divide[1,a]*FresnelF[Divide[a,Pi]] |
Failure | Failure | Skip | Successful |
| 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(exp(- a*t)*FresnelS(t), t = 0..infinity)=(1)/(a)*Fresnelg((a)/(Pi)) |
Integrate[Exp[- a*t]*FresnelS[t], {t, 0, Infinity}]=Divide[1,a]*FresnelG[Divide[a,Pi]] |
Failure | Failure | Skip | Successful |
| 7.17#Ex1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle y = \inverf@@{x}} | Error |
y = InverseErf[x] |
Error | Failure | - | Fail
DirectedInfinity[-1] <- {Rule[x, 1], Rule[y, 1]}DirectedInfinity[-1] <- {Rule[x, 1], Rule[y, 2]}DirectedInfinity[-1] <- {Rule[x, 1], Rule[y, 3]} |
| 7.17#Ex2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle y = \inverfc@@{x}} | Error |
y = InverseErfc[x] |
Error | Failure | - | Fail
1.0 <- {Rule[x, 1], Rule[y, 1]}2.0 <- {Rule[x, 1], Rule[y, 2]}3.0 <- {Rule[x, 1], Rule[y, 3]}DirectedInfinity[1] <- {Rule[x, 2], Rule[y, 1]}... skip entries to safe data |
| 7.18#Ex1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \repinterfc{-1}@{z} = \frac{2}{\sqrt{\pi}}e^{-z^{2}}} | erfc(- 1, z)=(2)/(sqrt(Pi))*exp(- (z)^(2)) |
I^(- 1)*Erfc[z]=Divide[2,Sqrt[Pi]]*Exp[- (z)^(2)] |
Successful | Failure | - | Fail
Complex[1.011483603950918, -0.8436484572858769] <- {Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[0.46363208502383696, 0.8642718813368542] <- {Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[0.46363208502383696, -2.864271881336854] <- {Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[1.011483603950918, -1.1563515427141229] <- {Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} |
| 7.18#Ex2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \repinterfc{0}@{z} = \erfc@@{z}} | erfc(0, z)= erfc(z) |
I^(0)*Erfc[z]= Erfc[z] |
Successful | Successful | - | - |
| 7.18.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \repinterfc{n}@{z} = \int_{z}^{\infty}\repinterfc{n-1}@{t}\diff{t}} | erfc(n, z)= int(erfc(n - 1, t), t = z..infinity) |
I^(n)*Erfc[z]= Integrate[I^(n - 1)*Erfc[t], {t, z, Infinity}] |
Failure | Failure | Skip | Fail
Complex[-0.30711932433427286, -0.06448523556221403] <- {Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[0.06448523556221401, -0.30711932433427286] <- {Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[0.30711932433427286, 0.06448523556221403] <- {Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[0.2407321945928082, 0.04386181151123664] <- {Rule[n, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 7.18.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \int_{z}^{\infty}\repinterfc{n-1}@{t}\diff{t} = \frac{2}{\sqrt{\pi}}\int_{z}^{\infty}\frac{(t-z)^{n}}{n!}e^{-t^{2}}\diff{t}} | int(erfc(n - 1, t), t = z..infinity)=(2)/(sqrt(Pi))*int(((t - z)^(n))/(factorial(n))*exp(- (t)^(2)), t = z..infinity) |
Integrate[I^(n - 1)*Erfc[t], {t, z, Infinity}]=Divide[2,Sqrt[Pi]]*Integrate[Divide[(t - z)^(n),(n)!]*Exp[- (t)^(2)], {t, z, Infinity}] |
Failure | Failure | Skip | Fail
Complex[-0.06643066657209085, 0.02648998567028575] <- {Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[-0.036107850584238765, -0.054264273946754926] <- {Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[0.04191638050136022, 0.039897144071178225] <- {Rule[n, 2], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[-0.03610785058423853, 0.054264273946755606] <- {Rule[n, 3], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 7.18.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \deriv{}{z}\repinterfc{n}@{z} = -\repinterfc{n-1}@{z}} | diff(erfc(n, z), z)= - erfc(n - 1, z) |
D[I^(n)*Erfc[z], z]= - I^(n - 1)*Erfc[z] |
Successful | Failure | - | Fail
Complex[-0.8436484572858769, -1.011483603950918] <- {Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[0.8642718813368542, -0.46363208502383696] <- {Rule[n, 3], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[-2.864271881336854, -0.46363208502383696] <- {Rule[n, 3], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[-1.1563515427141229, -1.011483603950918] <- {Rule[n, 3], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} |
| 7.18.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \deriv[n]{}{z}\left(e^{z^{2}}\erfc@@{z}\right) = (-1)^{n}2^{n}n!e^{z^{2}}\repinterfc{n}@{z}} | diff(exp((z)^(2))*erfc(z), [z$(n)])=(- 1)^(n)* (2)^(n)* factorial(n)*exp((z)^(2))*erfc(n, z) |
D[Exp[(z)^(2)]*Erfc[z], {z, n}]=(- 1)^(n)* (2)^(n)* (n)!*Exp[(z)^(2)]*I^(n)*Erfc[z] |
Failure | Failure | Successful | Fail
Complex[-8.131664243641417, -10.165585245606788] <- {Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[8.307941760049161, -10.383011529763138] <- {Rule[n, 3], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[-195.50543578827103, 111.66805229196896] <- {Rule[n, 3], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[-66.63896269283943, 34.38011968921443] <- {Rule[n, 3], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} |
| 7.18.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \deriv[2]{W}{z}+2z\deriv{W}{z}-2nW = 0} | diff(W, [z$(2)])+ 2*z*diff(W, z)- 2*n*W = 0 |
D[W, {z, 2}]+ 2*z*D[W, z]- 2*n*W = 0 |
Failure | Failure | Skip | Successful |
| 7.18.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \repinterfc{n}@{z} = \sum_{k=0}^{\infty}\frac{(-1)^{k}z^{k}}{2^{n-k}k!\EulerGamma@{1+\frac{1}{2}(n-k)}}} | erfc(n, z)= sum(((- 1)^(k)* (z)^(k))/((2)^(n - k)* factorial(k)*GAMMA(1 +(1)/(2)*(n - k))), k = 0..infinity) |
I^(n)*Erfc[z]= Sum[Divide[(- 1)^(k)* (z)^(k),(2)^(n - k)* (k)!*Gamma[1 +Divide[1,2]*(n - k)]], {k, 0, Infinity}] |
Failure | Failure | Skip | Fail
Complex[-0.3071193243342728, -0.06448523556221496] <- {Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[-0.0019454310098768, -0.280629338663987] <- {Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[0.2710114737500341, 0.01022096161545949] <- {Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[0.24073219459280773, 0.043861811511237594] <- {Rule[n, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 7.18.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \repinterfc{n}@{z} = -\frac{z}{n}\repinterfc{n-1}@{z}+\frac{1}{2n}\repinterfc{n-2}@{z}} | erfc(n, z)= -(z)/(n)*erfc(n - 1, z)+(1)/(2*n)*erfc(n - 2, z) |
I^(n)*Erfc[z]= -Divide[z,n]*I^(n - 1)*Erfc[z]+Divide[1,2*n]*I^(n - 2)*Erfc[z] |
Successful | Failure | - | Fail
Complex[0.45357088174560434, -0.11223852300991567] <- {Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[-0.18558922366932362, 0.1362991844027226] <- {Rule[n, 3], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}Complex[0.7572198179127398, -1.5268234761539925] <- {Rule[n, 3], Rule[z, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}Complex[1.3963799233276677, -3.163903851905481] <- {Rule[n, 3], Rule[z, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} |
| 7.18.E8 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (-1)^{n}\repinterfc{n}@{z}+\repinterfc{n}@{-z} = \frac{i^{-n}}{2^{n-1}n!}\HermitepolyH{n}@{iz}} | (- 1)^(n)* erfc(n, z)+ erfc(n, - z)=((I)^(- n))/((2)^(n - 1)* factorial(n))*HermiteH(n, I*z) |
(- 1)^(n)* I^(n)*Erfc[z]+ I^(n)*Erfc[- z]=Divide[(I)^(- n),(2)^(n - 1)* (n)!]*HermiteH[n, I*z] |
Failure | Failure | Successful | Fail
Complex[-2.280575605819109, -0.8078037006952131] <- {Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[-2.5, -4.0] <- {Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[0.6306597830504983, -4.613348288401651] <- {Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[-3.3762786436732717, 4.849050548797168] <- {Rule[n, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 7.18.E9 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \repinterfc{n}@{z} = e^{-z^{2}}\left(\frac{1}{2^{n}\EulerGamma@{\tfrac{1}{2}n+1}}\KummerconfhyperM@{\tfrac{1}{2}n+\tfrac{1}{2}}{\tfrac{1}{2}}{z^{2}}-\frac{z}{2^{n-1}\EulerGamma@{\tfrac{1}{2}n+\tfrac{1}{2}}}\KummerconfhyperM@{\tfrac{1}{2}n+1}{\tfrac{3}{2}}{z^{2}}\right)} | erfc(n, z)= exp(- (z)^(2))*((1)/((2)^(n)* GAMMA((1)/(2)*n + 1))*KummerM((1)/(2)*n +(1)/(2), (1)/(2), (z)^(2))-(z)/((2)^(n - 1)* GAMMA((1)/(2)*n +(1)/(2)))*KummerM((1)/(2)*n + 1, (3)/(2), (z)^(2))) |
I^(n)*Erfc[z]= Exp[- (z)^(2)]*(Divide[1,(2)^(n)* Gamma[Divide[1,2]*n + 1]]*Hypergeometric1F1[Divide[1,2]*n +Divide[1,2], Divide[1,2], (z)^(2)]-Divide[z,(2)^(n - 1)* Gamma[Divide[1,2]*n +Divide[1,2]]]*Hypergeometric1F1[Divide[1,2]*n + 1, Divide[3,2], (z)^(2)]) |
Failure | Failure | Successful | Fail
Complex[-0.3071193243342726, -0.06448523556221446] <- {Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[-0.0019454310098766699, -0.28062933866398737] <- {Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[0.27101147375003376, 0.010220961615459446] <- {Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[0.2407321945928081, 0.04386181151123732] <- {Rule[n, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 7.18.E10 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \repinterfc{n}@{z} = \frac{e^{-z^{2}}}{2^{n}\sqrt{\pi}}\KummerconfhyperU@{\tfrac{1}{2}n+\tfrac{1}{2}}{\tfrac{1}{2}}{z^{2}}} | erfc(n, z)=(exp(- (z)^(2)))/((2)^(n)*sqrt(Pi))*KummerU((1)/(2)*n +(1)/(2), (1)/(2), (z)^(2)) |
I^(n)*Erfc[z]=Divide[Exp[- (z)^(2)],(2)^(n)*Sqrt[Pi]]*HypergeometricU[Divide[1,2]*n +Divide[1,2], Divide[1,2], (z)^(2)] |
Failure | Failure | Fail 2.828427124+2.828427124*I <- {z = -2^(1/2)-I*2^(1/2), n = 1}.4754857140+3.986592840*I <- {z = -2^(1/2)-I*2^(1/2), n = 2}-1.178511301+2.592724863*I <- {z = -2^(1/2)-I*2^(1/2), n = 3}2.828427124-2.828427124*I <- {z = -2^(1/2)+I*2^(1/2), n = 1}... skip entries to safe data |
Fail
Complex[-0.30711932433427297, -0.06448523556221639] <- {Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[-0.0019454310098774158, -0.28062933866398587] <- {Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[0.2710114737500343, 0.010220961615459385] <- {Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[0.24073219459280773, 0.04386181151123868] <- {Rule[n, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 7.18.E11 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \repinterfc{n}@{z} = \frac{e^{-z^{2}/2}}{\sqrt{2^{n-1}\pi}}\paraU@{n+\tfrac{1}{2}}{z\sqrt{2}}} | erfc(n, z)=(exp(- (z)^(2)/ 2))/(sqrt((2)^(n - 1)* Pi))*CylinderU(n +(1)/(2), z*sqrt(2)) |
I^(n)*Erfc[z]=Divide[Exp[- (z)^(2)/ 2],Sqrt[(2)^(n - 1)* Pi]]*ParabolicCylinderD[-n +Divide[1,2] - 1/2, z*Sqrt[2]] |
Failure | Failure | Successful | Fail
Complex[-0.26663427796467404, -0.20400647408383285] <- {Rule[n, 1], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[-0.013159682786361896, -0.3122322253171296] <- {Rule[n, 2], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[0.2652586505052597, 0.005571565714632675] <- {Rule[n, 3], Rule[z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[0.281217240962407, 0.18338305003285552] <- {Rule[n, 1], Rule[z, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 7.20.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{\sigma\sqrt{2\pi}}\int_{-\infty}^{x}e^{-(t-m)^{2}/(2\sigma^{2})}\diff{t} = \frac{1}{2}\erfc@{\frac{m-x}{\sigma\sqrt{2}}}} | (1)/(sigma*sqrt(2*Pi))*int(exp(-(t - m)^(2)/(2*(sigma)^(2))), t = - infinity..x)=(1)/(2)*erfc((m - x)/(sigma*sqrt(2))) |
Divide[1,\[Sigma]*Sqrt[2*Pi]]*Integrate[Exp[-(t - m)^(2)/(2*(\[Sigma])^(2))], {t, - Infinity, x}]=Divide[1,2]*Erfc[Divide[m - x,\[Sigma]*Sqrt[2]]] |
Failure | Failure | Skip | Successful |
| 7.20.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{2}\erfc@{\frac{m-x}{\sigma\sqrt{2}}} = Q\left(\frac{m-x}{\sigma}\right)} | (1)/(2)*erfc((m - x)/(sigma*sqrt(2)))= Q*((m - x)/(sigma)) |
Divide[1,2]*Erfc[Divide[m - x,\[Sigma]*Sqrt[2]]]= Q*(Divide[m - x,\[Sigma]]) |
Failure | Failure | Fail .5000000000 <- {Q = 2^(1/2)+I*2^(1/2), sigma = 2^(1/2)+I*2^(1/2), m = 1, x = 1}1.646697588-.1349567498*I <- {Q = 2^(1/2)+I*2^(1/2), sigma = 2^(1/2)+I*2^(1/2), m = 1, x = 2}2.821306458-.2289406972*I <- {Q = 2^(1/2)+I*2^(1/2), sigma = 2^(1/2)+I*2^(1/2), m = 1, x = 3}-.6466975876+.1349567498*I <- {Q = 2^(1/2)+I*2^(1/2), sigma = 2^(1/2)+I*2^(1/2), m = 2, x = 1}... skip entries to safe data |
Fail
0.5 <- {Rule[m, 1], Rule[Q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[σ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[1.6466975876615073, -0.13495674973157074] <- {Rule[m, 1], Rule[Q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 2], Rule[σ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[2.8213064574274105, -0.22894069721759613] <- {Rule[m, 1], Rule[Q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 3], Rule[σ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}Complex[-0.6466975876615073, 0.13495674973157074] <- {Rule[m, 2], Rule[Q, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[x, 1], Rule[σ, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}... skip entries to safe data |
| 7.20.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle Q\left(\frac{m-x}{\sigma}\right) = P\left(\frac{x-m}{\sigma}\right)} | Q*((m - x)/(sigma))= P*((x - m)/(sigma)) |
Q*(Divide[m - x,\[Sigma]])= P*(Divide[x - m,\[Sigma]]) |
Failure | Failure | Skip | Skip |