Elliptic Integrals - 19.23 Integral Representations
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| DLMF | Formula | Constraints | Maple | Mathematica | Symbolic Maple |
Symbolic Mathematica |
Numeric Maple |
Numeric Mathematica |
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| 19.23.E1 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \CarlsonsymellintRF@{0}{y}{z} = \int_{0}^{\pi/2}(y\cos^{2}@@{\theta}+z\sin^{2}@@{\theta})^{-1/2}\diff{\theta}}
\CarlsonsymellintRF@{0}{y}{z} = \int_{0}^{\pi/2}(y\cos^{2}@@{\theta}+z\sin^{2}@@{\theta})^{-1/2}\diff{\theta} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | 0.5*int(1/(sqrt(t+0)*sqrt(t+y)*sqrt(t+x + y*I)), t = 0..infinity) = int((y*(cos(theta))^(2)+(x + y*I)*(sin(theta))^(2))^(- 1/2), theta = 0..Pi/2)
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EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]/Sqrt[x + y*I-0] == Integrate[(y*(Cos[\[Theta]])^(2)+(x + y*I)*(Sin[\[Theta]])^(2))^(- 1/2), {\[Theta], 0, Pi/2}, GenerateConditions->None]
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Aborted | Failure | Skipped - Because timed out | Failed [18 / 18]
Result: Complex[0.8397393007192011, 1.792316631638506]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}
Result: Complex[-1.057179647328743, -0.8381019542468571]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}
... skip entries to safe data |
| 19.23.E2 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \CarlsonsymellintRG@{0}{y}{z} = \frac{1}{2}\int_{0}^{\pi/2}(y\cos^{2}@@{\theta}+z\sin^{2}@@{\theta})^{1/2}\diff{\theta}}
\CarlsonsymellintRG@{0}{y}{z} = \frac{1}{2}\int_{0}^{\pi/2}(y\cos^{2}@@{\theta}+z\sin^{2}@@{\theta})^{1/2}\diff{\theta} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Error
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Sqrt[x + y*I-0]*(EllipticE[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]+(Cot[ArcCos[Sqrt[0/x + y*I]]])^2*EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]+Cot[ArcCos[Sqrt[0/x + y*I]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/x + y*I]]]^2]) == Divide[1,2]*Integrate[(y*(Cos[\[Theta]])^(2)+(x + y*I)*(Sin[\[Theta]])^(2))^(1/2), {\[Theta], 0, Pi/2}, GenerateConditions->None]
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Missing Macro Error | Failure | - | Failed [18 / 18]
Result: Plus[Complex[0.5014070071339144, -0.6068932953779227], Times[Complex[1.345607733249115, -0.5573689727459014], Plus[Complex[1.465481142300126, -0.24396122198922798], Times[Complex[0.2643318009908678, -0.8730286325904596], Power[Plus[1.0, Times[Complex[-1.0, -1.5], Power[k, 2]]], Rational[1, 2]]]]]]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}
Result: Plus[Complex[-0.9996439786591846, -0.22609983985234913], Times[Complex[1.345607733249115, 0.5573689727459014], Plus[Complex[1.0084590214609772, 0.7147093671486319], Times[Complex[0.2643318009908678, 0.8730286325904596], Power[Plus[1.0, Times[Complex[-1.0, 1.5], Power[k, 2]]], Rational[1, 2]]]]]]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}
... skip entries to safe data |
| 19.23.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \CarlsonsymellintRD@{0}{y}{z} = 3\int_{0}^{\pi/2}(y\cos^{2}@@{\theta}+z\sin^{2}@@{\theta})^{-3/2}\sin^{2}@@{\theta}\diff{\theta}}
\CarlsonsymellintRD@{0}{y}{z} = 3\int_{0}^{\pi/2}(y\cos^{2}@@{\theta}+z\sin^{2}@@{\theta})^{-3/2}\sin^{2}@@{\theta}\diff{\theta} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Error
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3*(EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]-EllipticE[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)])/((x + y*I-y)*(x + y*I-0)^(1/2)) == 3*Integrate[(y*(Cos[\[Theta]])^(2)+(x + y*I)*(Sin[\[Theta]])^(2))^(- 3/2)* (Sin[\[Theta]])^(2), {\[Theta], 0, Pi/2}, GenerateConditions->None]
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Missing Macro Error | Aborted | - | Skipped - Because timed out |
| 19.23.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \CarlsonsymellintRF@{0}{y}{z} = \frac{2}{\pi}\int_{0}^{\pi/2}\CarlsonellintRC@{y}{z\cos^{2}@@{\theta}}\diff{\theta}}
\CarlsonsymellintRF@{0}{y}{z} = \frac{2}{\pi}\int_{0}^{\pi/2}\CarlsonellintRC@{y}{z\cos^{2}@@{\theta}}\diff{\theta} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Error
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EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]/Sqrt[x + y*I-0] == Divide[2,Pi]*Integrate[1/Sqrt[(x + y*I)*(Cos[\[Theta]])^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-(y)/((x + y*I)*(Cos[\[Theta]])^(2))], {\[Theta], 0, Pi/2}, GenerateConditions->None]
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Missing Macro Error | Aborted | - | Skipped - Because timed out |
| 19.23.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{2}{\pi}\int_{0}^{\pi/2}\CarlsonellintRC@{y}{z\cos^{2}@@{\theta}}\diff{\theta} = \frac{2}{\pi}\int_{0}^{\infty}\CarlsonellintRC@{y\cosh^{2}@@{t}}{z}\diff{t}}
\frac{2}{\pi}\int_{0}^{\pi/2}\CarlsonellintRC@{y}{z\cos^{2}@@{\theta}}\diff{\theta} = \frac{2}{\pi}\int_{0}^{\infty}\CarlsonellintRC@{y\cosh^{2}@@{t}}{z}\diff{t} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Error
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Divide[2,Pi]*Integrate[1/Sqrt[(x + y*I)*(Cos[\[Theta]])^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-(y)/((x + y*I)*(Cos[\[Theta]])^(2))], {\[Theta], 0, Pi/2}, GenerateConditions->None] == Divide[2,Pi]*Integrate[1/Sqrt[x + y*I]*Hypergeometric2F1[1/2,1/2,3/2,1-(y*(Cosh[t])^(2))/(x + y*I)], {t, 0, Infinity}, GenerateConditions->None]
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Missing Macro Error | Aborted | - | Skipped - Because timed out |
| 19.23.E5 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \CarlsonsymellintRF@{x}{y}{z} = \frac{2}{\pi}\int_{0}^{\pi/2}\CarlsonellintRC@{x}{y\cos^{2}@@{\theta}+z\sin^{2}@@{\theta}}\diff{\theta}}
\CarlsonsymellintRF@{x}{y}{z} = \frac{2}{\pi}\int_{0}^{\pi/2}\CarlsonellintRC@{x}{y\cos^{2}@@{\theta}+z\sin^{2}@@{\theta}}\diff{\theta} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \realpart@@{y} > 0, \realpart@@{z} > 0} | Error
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EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x] == Divide[2,Pi]*Integrate[1/Sqrt[y*(Cos[\[Theta]])^(2)+(x + y*I)*(Sin[\[Theta]])^(2)]*Hypergeometric2F1[1/2,1/2,3/2,1-(x)/(y*(Cos[\[Theta]])^(2)+(x + y*I)*(Sin[\[Theta]])^(2))], {\[Theta], 0, Pi/2}, GenerateConditions->None]
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Missing Macro Error | Aborted | - | Skipped - Because timed out |
| 19.23.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 4\pi\CarlsonsymellintRF@{x}{y}{z} = \int_{0}^{2\pi}\!\!\!\!\int_{0}^{\pi}\frac{\sin@@{\theta}\diff{\theta}\diff{\phi}}{(x\sin^{2}@@{\theta}\cos^{2}@@{\phi}+y\sin^{2}@@{\theta}\sin^{2}@@{\phi}+z\cos^{2}@@{\theta})^{1/2}}}
4\pi\CarlsonsymellintRF@{x}{y}{z} = \int_{0}^{2\pi}\!\!\!\!\int_{0}^{\pi}\frac{\sin@@{\theta}\diff{\theta}\diff{\phi}}{(x\sin^{2}@@{\theta}\cos^{2}@@{\phi}+y\sin^{2}@@{\theta}\sin^{2}@@{\phi}+z\cos^{2}@@{\theta})^{1/2}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | 4*Pi*0.5*int(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+x + y*I)), t = 0..infinity) = int(int((sin(theta))/((x*(sin(theta))^(2)* (cos(phi))^(2)+ y*(sin(theta))^(2)* (sin(phi))^(2)+(x + y*I)*(cos(theta))^(2))^(1/2)), theta = 0..Pi), phi = 0..2*Pi)
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4*Pi*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x] == Integrate[Integrate[Divide[Sin[\[Theta]],(x*(Sin[\[Theta]])^(2)* (Cos[\[Phi]])^(2)+ y*(Sin[\[Theta]])^(2)* (Sin[\[Phi]])^(2)+(x + y*I)*(Cos[\[Theta]])^(2))^(1/2)], {\[Theta], 0, Pi}, GenerateConditions->None], {\[Phi], 0, 2*Pi}, GenerateConditions->None]
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Aborted | Aborted | Skipped - Because timed out | Skipped - Because timed out |
| 19.23.E7 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \CarlsonsymellintRG@{x}{y}{z} = \frac{1}{4}\int_{0}^{\infty}\frac{1}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}}\*\left(\frac{x}{t+x}+\frac{y}{t+y}+\frac{z}{t+z}\right)t\diff{t}}
\CarlsonsymellintRG@{x}{y}{z} = \frac{1}{4}\int_{0}^{\infty}\frac{1}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}}\*\left(\frac{x}{t+x}+\frac{y}{t+y}+\frac{z}{t+z}\right)t\diff{t} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | Error
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Sqrt[x + y*I-x]*(EllipticE[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]+(Cot[ArcCos[Sqrt[x/x + y*I]]])^2*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]+Cot[ArcCos[Sqrt[x/x + y*I]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[x/x + y*I]]]^2]) == Divide[1,4]*Integrate[Divide[1,Sqrt[t + x]*Sqrt[t + y]*Sqrt[t +(x + y*I)]]*(Divide[x,t + x]+Divide[y,t + y]+Divide[x + y*I,t +(x + y*I)])*t, {t, 0, Infinity}, GenerateConditions->None]
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Missing Macro Error | Aborted | - | Skipped - Because timed out |