Error Functions, Dawson’s and Fresnel Integrals - 7.5 Interrelations
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| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
|---|---|---|---|---|---|---|---|---|
| 7.5.E2 | 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)-\FresnelintF@{z}}
\Fresnelcosint@{z}+i\Fresnelsinint@{z} = \tfrac{1}{2}(1+i)-\FresnelintF@{z} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Error
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FresnelC[z]+ I*FresnelS[z] == Divide[1,2]*(1 + I)- (1+I)/2-FresnelC[z]-I*FresnelS[z]
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Missing Macro Error | Failure | - | Failed [7 / 7]
Result: Complex[1.0249430142401041, 0.8677085978643018]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Complex[-0.5229723981935741, 3.2881446840443265]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
... skip entries to safe data |
| 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}}}
\Fresnelcosint@{z} = \tfrac{1}{2}+\auxFresnelf@{z}\sin@{\tfrac{1}{2}\pi z^{2}}-\auxFresnelg@{z}\cos@{\tfrac{1}{2}\pi z^{2}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | FresnelC(z) = (1)/(2)+ Fresnelf(z)*sin((1)/(2)*Pi*(z)^(2))- Fresnelg(z)*cos((1)/(2)*Pi*(z)^(2))
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FresnelC[z] == Divide[1,2]+ FresnelF[z]*Sin[Divide[1,2]*Pi*(z)^(2)]- FresnelG[z]*Cos[Divide[1,2]*Pi*(z)^(2)]
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Successful | Failure | - | Successful [Tested: 7] |
| 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}}}
\Fresnelsinint@{z} = \tfrac{1}{2}-\auxFresnelf@{z}\cos@{\tfrac{1}{2}\pi z^{2}}-\auxFresnelg@{z}\sin@{\tfrac{1}{2}\pi z^{2}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | FresnelS(z) = (1)/(2)- Fresnelf(z)*cos((1)/(2)*Pi*(z)^(2))- Fresnelg(z)*sin((1)/(2)*Pi*(z)^(2))
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FresnelS[z] == Divide[1,2]- FresnelF[z]*Cos[Divide[1,2]*Pi*(z)^(2)]- FresnelG[z]*Sin[Divide[1,2]*Pi*(z)^(2)]
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Successful | Failure | - | Successful [Tested: 7] |
| 7.5.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle e^{-\frac{1}{2}\pi iz^{2}}\FresnelintF@{z} = \auxFresnelg@{z}+i\auxFresnelf@{z}}
e^{-\frac{1}{2}\pi iz^{2}}\FresnelintF@{z} = \auxFresnelg@{z}+i\auxFresnelf@{z} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Error
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Exp[-Divide[1,2]*Pi*I*(z)^(2)]*(1+I)/2-FresnelC[z]-I*FresnelS[z] == FresnelG[z]+ I*FresnelF[z]
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Missing Macro Error | Failure | - | Failed [6 / 7]
Result: Complex[2.0955908860316255, -0.6505223669676224]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Complex[-0.08422623998042833, -1.3932392044453867]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
... skip entries to safe data |
| 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})}
e^{+\frac{1}{2}\pi iz^{2}}(\auxFresnelg@{z}+ i\auxFresnelf@{z}) = \tfrac{1}{2}(1+ i)-(\Fresnelcosint@{z}+ i\Fresnelsinint@{z}) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | exp(+(1)/(2)*Pi*I*(z)^(2))*(Fresnelg(z)+ I*Fresnelf(z)) = (1)/(2)*(1 + I)-(FresnelC(z)+ I*FresnelS(z))
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Exp[+Divide[1,2]*Pi*I*(z)^(2)]*(FresnelG[z]+ I*FresnelF[z]) == Divide[1,2]*(1 + I)-(FresnelC[z]+ I*FresnelS[z])
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Successful | Failure | - | Successful [Tested: 7] |
| 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})}
e^{-\frac{1}{2}\pi iz^{2}}(\auxFresnelg@{z}- i\auxFresnelf@{z}) = \tfrac{1}{2}(1- i)-(\Fresnelcosint@{z}- i\Fresnelsinint@{z}) |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | exp(-(1)/(2)*Pi*I*(z)^(2))*(Fresnelg(z)- I*Fresnelf(z)) = (1)/(2)*(1 - I)-(FresnelC(z)- I*FresnelS(z))
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Exp[-Divide[1,2]*Pi*I*(z)^(2)]*(FresnelG[z]- I*FresnelF[z]) == Divide[1,2]*(1 - I)-(FresnelC[z]- I*FresnelS[z])
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Successful | Failure | - | Successful [Tested: 7] |
| 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}}
\Fresnelcosint@{z}+ i\Fresnelsinint@{z} = \tfrac{1}{2}(1+ i)\erf@@{\zeta} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | FresnelC(z)+ I*FresnelS(z) = (1)/(2)*(1 + I)*erf((1)/(2)*sqrt(Pi)*(1 - I)*z)
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FresnelC[z]+ I*FresnelS[z] == Divide[1,2]*(1 + I)*Erf[Divide[1,2]*Sqrt[Pi]*(1 - I)*z]
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Successful | Successful | Skip - symbolical successful subtest | Successful [Tested: 7] |
| 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}}
\Fresnelcosint@{z}- i\Fresnelsinint@{z} = \tfrac{1}{2}(1- i)\erf@@{\zeta} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | FresnelC(z)- I*FresnelS(z) = (1)/(2)*(1 - I)*erf((1)/(2)*sqrt(Pi)*(1 - I)*z)
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FresnelC[z]- I*FresnelS[z] == Divide[1,2]*(1 - I)*Erf[Divide[1,2]*Sqrt[Pi]*(1 - I)*z]
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Failure | Failure | Failed [7 / 7] Result: 1.210218044+.7739577054*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}
Result: -2.077926642+.2509853077*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}
... skip entries to safe data |
Failed [7 / 7]
Result: Complex[1.210218043090013, 0.7739577062168396]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Complex[-2.0779266409543133, 0.2509853080232649]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
... skip entries to safe data |
| 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}}
\auxFresnelg@{z}+ i\auxFresnelf@{z} = \tfrac{1}{2}(1+ i)e^{\zeta^{2}}\erfc@@{\zeta} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Fresnelg(z)+ I*Fresnelf(z) = (1)/(2)*(1 + I)*exp(((1)/(2)*sqrt(Pi)*(1 - I)*z)^(2))*erfc((1)/(2)*sqrt(Pi)*(1 - I)*z)
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FresnelG[z]+ I*FresnelF[z] == Divide[1,2]*(1 + I)*Exp[(Divide[1,2]*Sqrt[Pi]*(1 - I)*z)^(2)]*Erfc[Divide[1,2]*Sqrt[Pi]*(1 - I)*z]
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Successful | Failure | Skip - symbolical successful subtest | Successful [Tested: 7] |
| 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}}
\auxFresnelg@{z}- i\auxFresnelf@{z} = \tfrac{1}{2}(1- i)e^{\zeta^{2}}\erfc@@{\zeta} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Fresnelg(z)- I*Fresnelf(z) = (1)/(2)*(1 - I)*exp(((1)/(2)*sqrt(Pi)*(1 - I)*z)^(2))*erfc((1)/(2)*sqrt(Pi)*(1 - I)*z)
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FresnelG[z]- I*FresnelF[z] == Divide[1,2]*(1 - I)*Exp[(Divide[1,2]*Sqrt[Pi]*(1 - I)*z)^(2)]*Erfc[Divide[1,2]*Sqrt[Pi]*(1 - I)*z]
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Failure | Failure | Failed [7 / 7] Result: -.2860780540-.1977870141*I
Test Values: {z = 1/2*3^(1/2)+1/2*I}
Result: -.1472580850-5.018337775*I
Test Values: {z = -1/2+1/2*I*3^(1/2)}
... skip entries to safe data |
Failed [7 / 7]
Result: Complex[-0.2860780524436176, -0.19778701442673574]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Result: Complex[-0.14725808362732817, -5.018337771876615]
Test Values: {Rule[z, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
... skip entries to safe data |
| 7.5.E11 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\FresnelintF@{x}|^{2} = \auxFresnelf^{2}@{x}+\auxFresnelg^{2}@{x}}
|\FresnelintF@{x}|^{2} = \auxFresnelf^{2}@{x}+\auxFresnelg^{2}@{x} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x \geq 0} | Error
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(Abs[(1+I)/2-FresnelC[x]-I*FresnelS[x]])^(2) == (FresnelF[x])^(2)+ (FresnelG[x])^(2)
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Missing Macro Error | Failure | - | Successful [Tested: 3] |
| 7.5.E12 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle |\FresnelintF@{x}|^{2} = 2+\auxFresnelf^{2}@{-x}+\auxFresnelg^{2}@{-x}-2\sqrt{2}\cos@{\tfrac{1}{4}\pi+\tfrac{1}{2}\pi x^{2}}\auxFresnelf@{-x}-2\sqrt{2}\cos@{\tfrac{1}{4}\pi-\tfrac{1}{2}\pi x^{2}}\auxFresnelg@{-x}}
|\FresnelintF@{x}|^{2} = 2+\auxFresnelf^{2}@{-x}+\auxFresnelg^{2}@{-x}-2\sqrt{2}\cos@{\tfrac{1}{4}\pi+\tfrac{1}{2}\pi x^{2}}\auxFresnelf@{-x}-2\sqrt{2}\cos@{\tfrac{1}{4}\pi-\tfrac{1}{2}\pi x^{2}}\auxFresnelg@{-x} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x \leq 0} | Error
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(Abs[(1+I)/2-FresnelC[x]-I*FresnelS[x]])^(2) == 2 + (FresnelF[- x])^(2)+ (FresnelG[- x])^(2)- 2*Sqrt[2]*Cos[Divide[1,4]*Pi +Divide[1,2]*Pi*(x)^(2)]*FresnelF[- x]- 2*Sqrt[2]*Cos[Divide[1,4]*Pi -Divide[1,2]*Pi*(x)^(2)]*FresnelG[- x]
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Missing Macro Error | Failure | - | Skip - No test values generated |