Elliptic Integrals - 19.23 Integral Representations

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
19.23.E1 R F ⁡ ( 0 , y , z ) = ∫ 0 π / 2 ( y ⁢ cos 2 ⁡ θ + z ⁢ sin 2 ⁡ θ ) - 1 / 2 ⁢ d θ Carlson-integral-RF 0 𝑦 𝑧 superscript subscript 0 𝜋 2 superscript 𝑦 2 𝜃 𝑧 2 𝜃 1 2 𝜃 {\displaystyle{\displaystyle R_{F}\left(0,y,z\right)=\int_{0}^{\pi/2}(y{\cos^{% 2}}\theta+z{\sin^{2}}\theta)^{-1/2}\mathrm{d}\theta}}
\CarlsonsymellintRF@{0}{y}{z} = \int_{0}^{\pi/2}(y\cos^{2}@@{\theta}+z\sin^{2}@@{\theta})^{-1/2}\diff{\theta}		 

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)

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]
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 R G ⁡ ( 0 , y , z ) = 1 2 ⁢ ∫ 0 π / 2 ( y ⁢ cos 2 ⁡ θ + z ⁢ sin 2 ⁡ θ ) 1 / 2 ⁢ d θ Carlson-integral-RG 0 𝑦 𝑧 1 2 superscript subscript 0 𝜋 2 superscript 𝑦 2 𝜃 𝑧 2 𝜃 1 2 𝜃 {\displaystyle{\displaystyle R_{G}\left(0,y,z\right)=\frac{1}{2}\int_{0}^{\pi/% 2}(y{\cos^{2}}\theta+z{\sin^{2}}\theta)^{1/2}\mathrm{d}\theta}}
\CarlsonsymellintRG@{0}{y}{z} = \frac{1}{2}\int_{0}^{\pi/2}(y\cos^{2}@@{\theta}+z\sin^{2}@@{\theta})^{1/2}\diff{\theta}		 

Error

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]
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 R D ⁡ ( 0 , y , z ) = 3 ⁢ ∫ 0 π / 2 ( y ⁢ cos 2 ⁡ θ + z ⁢ sin 2 ⁡ θ ) - 3 / 2 ⁢ sin 2 ⁡ θ ⁢ d θ Carlson-integral-RD 0 𝑦 𝑧 3 superscript subscript 0 𝜋 2 superscript 𝑦 2 𝜃 𝑧 2 𝜃 3 2 2 𝜃 𝜃 {\displaystyle{\displaystyle R_{D}\left(0,y,z\right)=3\int_{0}^{\pi/2}(y{\cos^% {2}}\theta+z{\sin^{2}}\theta)^{-3/2}{\sin^{2}}\theta\mathrm{d}\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}		 

Error

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]
Missing Macro Error Aborted - Skipped - Because timed out
19.23.E4 R F ⁡ ( 0 , y , z ) = 2 π ⁢ ∫ 0 π / 2 R C ⁡ ( y , z ⁢ cos 2 ⁡ θ ) ⁢ d θ Carlson-integral-RF 0 𝑦 𝑧 2 𝜋 superscript subscript 0 𝜋 2 Carlson-integral-RC 𝑦 𝑧 2 𝜃 𝜃 {\displaystyle{\displaystyle R_{F}\left(0,y,z\right)=\frac{2}{\pi}\int_{0}^{% \pi/2}R_{C}\left(y,z{\cos^{2}}\theta\right)\mathrm{d}\theta}}
\CarlsonsymellintRF@{0}{y}{z} = \frac{2}{\pi}\int_{0}^{\pi/2}\CarlsonellintRC@{y}{z\cos^{2}@@{\theta}}\diff{\theta}		 

Error

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]
Missing Macro Error Aborted - Skipped - Because timed out
19.23.E4 2 π ⁢ ∫ 0 π / 2 R C ⁡ ( y , z ⁢ cos 2 ⁡ θ ) ⁢ d θ = 2 π ⁢ ∫ 0 ∞ R C ⁡ ( y ⁢ cosh 2 ⁡ t , z ) ⁢ d t 2 𝜋 superscript subscript 0 𝜋 2 Carlson-integral-RC 𝑦 𝑧 2 𝜃 𝜃 2 𝜋 superscript subscript 0 Carlson-integral-RC 𝑦 2 𝑡 𝑧 𝑡 {\displaystyle{\displaystyle\frac{2}{\pi}\int_{0}^{\pi/2}R_{C}\left(y,z{\cos^{% 2}}\theta\right)\mathrm{d}\theta=\frac{2}{\pi}\int_{0}^{\infty}R_{C}\left(y{% \cosh^{2}}t,z\right)\mathrm{d}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}		 

Error

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]
Missing Macro Error Aborted - Skipped - Because timed out
19.23.E5 R F ⁡ ( x , y , z ) = 2 π ⁢ ∫ 0 π / 2 R C ⁡ ( x , y ⁢ cos 2 ⁡ θ + z ⁢ sin 2 ⁡ θ ) ⁢ d θ Carlson-integral-RF 𝑥 𝑦 𝑧 2 𝜋 superscript subscript 0 𝜋 2 Carlson-integral-RC 𝑥 𝑦 2 𝜃 𝑧 2 𝜃 𝜃 {\displaystyle{\displaystyle R_{F}\left(x,y,z\right)=\frac{2}{\pi}\int_{0}^{% \pi/2}R_{C}\left(x,y{\cos^{2}}\theta+z{\sin^{2}}\theta\right)\mathrm{d}\theta}}
\CarlsonsymellintRF@{x}{y}{z} = \frac{2}{\pi}\int_{0}^{\pi/2}\CarlsonellintRC@{x}{y\cos^{2}@@{\theta}+z\sin^{2}@@{\theta}}\diff{\theta}		 ℜ ⁡ y > 0 , ℜ ⁡ z > 0 formulae-sequence 𝑦 0 𝑧 0 {\displaystyle{\displaystyle\Re y>0,\Re z>0}} 
Error

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]
Missing Macro Error Aborted - Skipped - Because timed out
19.23.E6 4 ⁢ π ⁢ R F ⁡ ( x , y , z ) = ∫ 0 2 ⁢ π ∫ 0 π sin ⁡ θ ⁢ d θ ⁢ d ϕ ( x ⁢ sin 2 ⁡ θ ⁢ cos 2 ⁡ ϕ + y ⁢ sin 2 ⁡ θ ⁢ sin 2 ⁡ ϕ + z ⁢ cos 2 ⁡ θ ) 1 / 2 4 𝜋 Carlson-integral-RF 𝑥 𝑦 𝑧 superscript subscript 0 2 𝜋 superscript subscript 0 𝜋 𝜃 𝜃 italic-ϕ superscript 𝑥 2 𝜃 2 italic-ϕ 𝑦 2 𝜃 2 italic-ϕ 𝑧 2 𝜃 1 2 {\displaystyle{\displaystyle 4\pi R_{F}\left(x,y,z\right)=\int_{0}^{2\pi}\!\!% \!\!\int_{0}^{\pi}\frac{\sin\theta\mathrm{d}\theta\mathrm{d}\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}}		 

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)

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]
Aborted Aborted Skipped - Because timed out Skipped - Because timed out
19.23.E7 R G ⁡ ( x , y , z ) = 1 4 ⁢ ∫ 0 ∞ 1 t + x ⁢ t + y ⁢ t + z ⁢ ( x t + x + y t + y + z t + z ) ⁢ t ⁢ d t Carlson-integral-RG 𝑥 𝑦 𝑧 1 4 superscript subscript 0 1 𝑡 𝑥 𝑡 𝑦 𝑡 𝑧 𝑥 𝑡 𝑥 𝑦 𝑡 𝑦 𝑧 𝑡 𝑧 𝑡 𝑡 {\displaystyle{\displaystyle R_{G}\left(x,y,z\right)=\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\mathrm{d}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}		 

Error

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]
Missing Macro Error Aborted - Skipped - Because timed out
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