Elliptic Integrals - 19.28 Integrals of Elliptic Integrals

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DLMF Formula Constraints Maple Mathematica Symbolic
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19.28.E1 ∫ 0 1 t σ - 1 ⁢ R F ⁡ ( 0 , t , 1 ) ⁢ d t = 1 2 ⁢ ( B ⁡ ( σ , 1 2 ) ) 2 superscript subscript 0 1 superscript 𝑡 𝜎 1 Carlson-integral-RF 0 𝑡 1 𝑡 1 2 superscript Euler-Beta 𝜎 1 2 2 {\displaystyle{\displaystyle\int_{0}^{1}t^{\sigma-1}R_{F}\left(0,t,1\right)% \mathrm{d}t=\tfrac{1}{2}\left(\mathrm{B}\left(\sigma,\tfrac{1}{2}\right)\right% )^{2}}}
\int_{0}^{1}t^{\sigma-1}\CarlsonsymellintRF@{0}{t}{1}\diff{t} = \tfrac{1}{2}\left(\EulerBeta@{\sigma}{\tfrac{1}{2}}\right)^{2}		 ℜ ⁡ ( σ ) > 0 , ℜ ⁡ ( ( σ ) + b ) > 0 , ℜ ⁡ ( a + ( 1 2 ) ) > 0 formulae-sequence 𝜎 0 formulae-sequence 𝜎 𝑏 0 𝑎 1 2 0 {\displaystyle{\displaystyle\Re(\sigma)>0,\Re((\sigma)+b)>0,\Re(a+(\tfrac{1}{2% }))>0}} 
int((t)^(sigma - 1)* 0.5*int(1/(sqrt(t+0)*sqrt(t+t)*sqrt(t+1)), t = 0..infinity), t = 0..1) = (1)/(2)*(Beta(sigma, (1)/(2)))^(2)

Integrate[(t)^(\[Sigma]- 1)* EllipticF[ArcCos[Sqrt[0/1]],(1-t)/(1-0)]/Sqrt[1-0], {t, 0, 1}, GenerateConditions->None] == Divide[1,2]*(Beta[\[Sigma], Divide[1,2]])^(2)
Failure Aborted
Failed [10 / 10]
Result: Float(undefined)+1.162857938*I
Test Values: {sigma = 1/2*3^(1/2)+1/2*I}


Result: Float(undefined)+.9984297790*I
Test Values: {sigma = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
 Skipped - Because timed out
19.28.E2 ∫ 0 1 t σ - 1 ⁢ R G ⁡ ( 0 , t , 1 ) ⁢ d t = σ 4 ⁢ σ + 2 ⁢ ( B ⁡ ( σ , 1 2 ) ) 2 superscript subscript 0 1 superscript 𝑡 𝜎 1 Carlson-integral-RG 0 𝑡 1 𝑡 𝜎 4 𝜎 2 superscript Euler-Beta 𝜎 1 2 2 {\displaystyle{\displaystyle\int_{0}^{1}t^{\sigma-1}R_{G}\left(0,t,1\right)% \mathrm{d}t=\frac{\sigma}{4\sigma+2}\left(\mathrm{B}\left(\sigma,\tfrac{1}{2}% \right)\right)^{2}}}
\int_{0}^{1}t^{\sigma-1}\CarlsonsymellintRG@{0}{t}{1}\diff{t} = \frac{\sigma}{4\sigma+2}\left(\EulerBeta@{\sigma}{\tfrac{1}{2}}\right)^{2}		 ℜ ⁡ ( σ ) > 0 , ℜ ⁡ ( ( σ ) + b ) > 0 , ℜ ⁡ ( a + ( 1 2 ) ) > 0 formulae-sequence 𝜎 0 formulae-sequence 𝜎 𝑏 0 𝑎 1 2 0 {\displaystyle{\displaystyle\Re(\sigma)>0,\Re((\sigma)+b)>0,\Re(a+(\tfrac{1}{2% }))>0}} 
Error

Integrate[(t)^(\[Sigma]- 1)* Sqrt[1-0]*(EllipticE[ArcCos[Sqrt[0/1]],(1-t)/(1-0)]+(Cot[ArcCos[Sqrt[0/1]]])^2*EllipticF[ArcCos[Sqrt[0/1]],(1-t)/(1-0)]+Cot[ArcCos[Sqrt[0/1]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[0/1]]]^2]), {t, 0, 1}, GenerateConditions->None] == Divide[\[Sigma],4*\[Sigma]+ 2]*(Beta[\[Sigma], Divide[1,2]])^(2)
Missing Macro Error Aborted - Skipped - Because timed out
19.28.E3 ∫ 0 1 t σ - 1 ⁢ ( 1 - t ) ⁢ R D ⁡ ( 0 , t , 1 ) ⁢ d t = 3 4 ⁢ σ + 2 ⁢ ( B ⁡ ( σ , 1 2 ) ) 2 superscript subscript 0 1 superscript 𝑡 𝜎 1 1 𝑡 Carlson-integral-RD 0 𝑡 1 𝑡 3 4 𝜎 2 superscript Euler-Beta 𝜎 1 2 2 {\displaystyle{\displaystyle\int_{0}^{1}t^{\sigma-1}(1-t)R_{D}\left(0,t,1% \right)\mathrm{d}t=\frac{3}{4\sigma+2}\left(\mathrm{B}\left(\sigma,\tfrac{1}{2% }\right)\right)^{2}}}
\int_{0}^{1}t^{\sigma-1}(1-t)\CarlsonsymellintRD@{0}{t}{1}\diff{t} = \frac{3}{4\sigma+2}\left(\EulerBeta@{\sigma}{\tfrac{1}{2}}\right)^{2}		 ℜ ⁡ ( σ ) > 0 , ℜ ⁡ ( ( σ ) + b ) > 0 , ℜ ⁡ ( a + ( 1 2 ) ) > 0 formulae-sequence 𝜎 0 formulae-sequence 𝜎 𝑏 0 𝑎 1 2 0 {\displaystyle{\displaystyle\Re(\sigma)>0,\Re((\sigma)+b)>0,\Re(a+(\tfrac{1}{2% }))>0}} 
Error

Integrate[(t)^(\[Sigma]- 1)*(1 - t)*3*(EllipticF[ArcCos[Sqrt[0/1]],(1-t)/(1-0)]-EllipticE[ArcCos[Sqrt[0/1]],(1-t)/(1-0)])/((1-t)*(1-0)^(1/2)), {t, 0, 1}, GenerateConditions->None] == Divide[3,4*\[Sigma]+ 2]*(Beta[\[Sigma], Divide[1,2]])^(2)
Missing Macro Error Aborted - Skipped - Because timed out
19.28.E5 ∫ z ∞ R D ⁡ ( x , y , t ) ⁢ d t = 6 ⁢ R F ⁡ ( x , y , z ) superscript subscript 𝑧 Carlson-integral-RD 𝑥 𝑦 𝑡 𝑡 6 Carlson-integral-RF 𝑥 𝑦 𝑧 {\displaystyle{\displaystyle\int_{z}^{\infty}R_{D}\left(x,y,t\right)\mathrm{d}% t=6R_{F}\left(x,y,z\right)}}
\int_{z}^{\infty}\CarlsonsymellintRD@{x}{y}{t}\diff{t} = 6\CarlsonsymellintRF@{x}{y}{z}		 

Error

Integrate[3*(EllipticF[ArcCos[Sqrt[x/t]],(t-y)/(t-x)]-EllipticE[ArcCos[Sqrt[x/t]],(t-y)/(t-x)])/((t-y)*(t-x)^(1/2)), {t, (x + y*I), Infinity}, GenerateConditions->None] == 6*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x]
Missing Macro Error Aborted - Skipped - Because timed out
19.28.E6 ∫ 0 1 R D ⁡ ( x , y , v 2 ⁢ z + ( 1 - v 2 ) ⁢ p ) ⁢ d v = R J ⁡ ( x , y , z , p ) superscript subscript 0 1 Carlson-integral-RD 𝑥 𝑦 superscript 𝑣 2 𝑧 1 superscript 𝑣 2 𝑝 𝑣 Carlson-integral-RJ 𝑥 𝑦 𝑧 𝑝 {\displaystyle{\displaystyle\int_{0}^{1}R_{D}\left(x,y,v^{2}z+(1-v^{2})p\right% )\mathrm{d}v=R_{J}\left(x,y,z,p\right)}}
\int_{0}^{1}\CarlsonsymellintRD@{x}{y}{v^{2}z+(1-v^{2})p}\diff{v} = \CarlsonsymellintRJ@{x}{y}{z}{p}		 

Error

Integrate[3*(EllipticF[ArcCos[Sqrt[x/(v)^(2)*(x + y*I)+(1 - (v)^(2))*p]],((v)^(2)*(x + y*I)+(1 - (v)^(2))*p-y)/((v)^(2)*(x + y*I)+(1 - (v)^(2))*p-x)]-EllipticE[ArcCos[Sqrt[x/(v)^(2)*(x + y*I)+(1 - (v)^(2))*p]],((v)^(2)*(x + y*I)+(1 - (v)^(2))*p-y)/((v)^(2)*(x + y*I)+(1 - (v)^(2))*p-x)])/(((v)^(2)*(x + y*I)+(1 - (v)^(2))*p-y)*((v)^(2)*(x + y*I)+(1 - (v)^(2))*p-x)^(1/2)), {v, 0, 1}, GenerateConditions->None] == 3*(x + y*I-x)/(x + y*I-p)*(EllipticPi[(x + y*I-p)/(x + y*I-x),ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/Sqrt[x + y*I-x]
Missing Macro Error Aborted - Skipped - Because timed out
19.28.E7 ∫ 0 ∞ R J ⁡ ( x , y , z , r 2 ) ⁢ d r = 3 2 ⁢ π ⁢ R F ⁡ ( x ⁢ y , x ⁢ z , y ⁢ z ) superscript subscript 0 Carlson-integral-RJ 𝑥 𝑦 𝑧 superscript 𝑟 2 𝑟 3 2 𝜋 Carlson-integral-RF 𝑥 𝑦 𝑥 𝑧 𝑦 𝑧 {\displaystyle{\displaystyle\int_{0}^{\infty}R_{J}\left(x,y,z,r^{2}\right)% \mathrm{d}r=\tfrac{3}{2}\pi R_{F}\left(xy,xz,yz\right)}}
\int_{0}^{\infty}\CarlsonsymellintRJ@{x}{y}{z}{r^{2}}\diff{r} = \tfrac{3}{2}\pi\CarlsonsymellintRF@{xy}{xz}{yz}		 

Error

Integrate[3*(x + y*I-x)/(x + y*I-(r)^(2))*(EllipticPi[(x + y*I-(r)^(2))/(x + y*I-x),ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/Sqrt[x + y*I-x], {r, 0, Infinity}, GenerateConditions->None] == Divide[3,2]*Pi*EllipticF[ArcCos[Sqrt[x*y/y*(x + y*I)]],(y*(x + y*I)-x*(x + y*I))/(y*(x + y*I)-x*y)]/Sqrt[y*(x + y*I)-x*y]
Missing Macro Error Aborted - Skipped - Because timed out
19.28.E8 ∫ 0 ∞ R J ⁡ ( t ⁢ x , y , z , t ⁢ p ) ⁢ d t = 6 p ⁢ R C ⁡ ( p , x ) ⁢ R F ⁡ ( 0 , y , z ) superscript subscript 0 Carlson-integral-RJ 𝑡 𝑥 𝑦 𝑧 𝑡 𝑝 𝑡 6 𝑝 Carlson-integral-RC 𝑝 𝑥 Carlson-integral-RF 0 𝑦 𝑧 {\displaystyle{\displaystyle\int_{0}^{\infty}R_{J}\left(tx,y,z,tp\right)% \mathrm{d}t=\frac{6}{\sqrt{p}}R_{C}\left(p,x\right)R_{F}\left(0,y,z\right)}}
\int_{0}^{\infty}\CarlsonsymellintRJ@{tx}{y}{z}{tp}\diff{t} = \frac{6}{\sqrt{p}}\CarlsonellintRC@{p}{x}\CarlsonsymellintRF@{0}{y}{z}		 

Error

Integrate[3*(x + y*I-t*x)/(x + y*I-t*p)*(EllipticPi[(x + y*I-t*p)/(x + y*I-t*x),ArcCos[Sqrt[t*x/x + y*I]],(x + y*I-y)/(x + y*I-t*x)]-EllipticF[ArcCos[Sqrt[t*x/x + y*I]],(x + y*I-y)/(x + y*I-t*x)])/Sqrt[x + y*I-t*x], {t, 0, Infinity}, GenerateConditions->None] == Divide[6,Sqrt[p]]*1/Sqrt[x]*Hypergeometric2F1[1/2,1/2,3/2,1-(p)/(x)]*EllipticF[ArcCos[Sqrt[0/x + y*I]],(x + y*I-y)/(x + y*I-0)]/Sqrt[x + y*I-0]
Missing Macro Error Aborted - Skipped - Because timed out
19.28.E9 ∫ 0 π / 2 R F ⁡ ( sin 2 ⁡ θ ⁢ cos 2 ⁡ ( x + y ) , sin 2 ⁡ θ ⁢ cos 2 ⁡ ( x - y ) , 1 ) ⁢ d θ = R F ⁡ ( 0 , cos 2 ⁡ x , 1 ) ⁢ R F ⁡ ( 0 , cos 2 ⁡ y , 1 ) superscript subscript 0 𝜋 2 Carlson-integral-RF 2 𝜃 2 𝑥 𝑦 2 𝜃 2 𝑥 𝑦 1 𝜃 Carlson-integral-RF 0 2 𝑥 1 Carlson-integral-RF 0 2 𝑦 1 {\displaystyle{\displaystyle\int_{0}^{\pi/2}R_{F}\left({\sin^{2}}\theta{\cos^{% 2}}\left(x+y\right),{\sin^{2}}\theta{\cos^{2}}\left(x-y\right),1\right)\mathrm% {d}\theta=R_{F}\left(0,{\cos^{2}}x,1\right)R_{F}\left(0,{\cos^{2}}y,1\right)}}
\int_{0}^{\pi/2}\CarlsonsymellintRF@{\sin^{2}@@{\theta}\cos^{2}@{x+y}}{\sin^{2}@@{\theta}\cos^{2}@{x-y}}{1}\diff{\theta} = \CarlsonsymellintRF@{0}{\cos^{2}@@{x}}{1}\CarlsonsymellintRF@{0}{\cos^{2}@@{y}}{1}		 

int(0.5*int(1/(sqrt(t+(sin(theta))^(2)* (cos(x + y))^(2))*sqrt(t+(sin(theta))^(2)* (cos(x - y))^(2))*sqrt(t+1)), t = 0..infinity), theta = 0..Pi/2) = 0.5*int(1/(sqrt(t+0)*sqrt(t+(cos(x))^(2))*sqrt(t+1)), t = 0..infinity)*0.5*int(1/(sqrt(t+0)*sqrt(t+(cos(y))^(2))*sqrt(t+1)), t = 0..infinity)

Integrate[EllipticF[ArcCos[Sqrt[(Sin[\[Theta]])^(2)* (Cos[x + y])^(2)/1]],(1-(Sin[\[Theta]])^(2)* (Cos[x - y])^(2))/(1-(Sin[\[Theta]])^(2)* (Cos[x + y])^(2))]/Sqrt[1-(Sin[\[Theta]])^(2)* (Cos[x + y])^(2)], {\[Theta], 0, Pi/2}, GenerateConditions->None] == EllipticF[ArcCos[Sqrt[0/1]],(1-(Cos[x])^(2))/(1-0)]/Sqrt[1-0]*EllipticF[ArcCos[Sqrt[0/1]],(1-(Cos[y])^(2))/(1-0)]/Sqrt[1-0]
Aborted Aborted Skipped - Because timed out Skipped - Because timed out
19.28.E10 ∫ 0 ∞ R F ⁡ ( ( a ⁢ c + b ⁢ d ) 2 , ( a ⁢ d + b ⁢ c ) 2 , 4 ⁢ a ⁢ b ⁢ c ⁢ d ⁢ cosh 2 ⁡ z ) ⁢ d z = 1 2 ⁢ R F ⁡ ( 0 , a 2 , b 2 ) ⁢ R F ⁡ ( 0 , c 2 , d 2 ) superscript subscript 0 Carlson-integral-RF superscript 𝑎 𝑐 𝑏 𝑑 2 superscript 𝑎 𝑑 𝑏 𝑐 2 4 𝑎 𝑏 𝑐 𝑑 2 𝑧 𝑧 1 2 Carlson-integral-RF 0 superscript 𝑎 2 superscript 𝑏 2 Carlson-integral-RF 0 superscript 𝑐 2 superscript 𝑑 2 {\displaystyle{\displaystyle\int_{0}^{\infty}R_{F}\left((ac+bd)^{2},(ad+bc)^{2% },4abcd{\cosh^{2}}z\right)\mathrm{d}z=\tfrac{1}{2}R_{F}\left(0,a^{2},b^{2}% \right)R_{F}\left(0,c^{2},d^{2}\right)}}
\int_{0}^{\infty}\CarlsonsymellintRF@{(ac+bd)^{2}}{(ad+bc)^{2}}{4abcd\cosh^{2}@@{z}}\diff{z} = \tfrac{1}{2}\CarlsonsymellintRF@{0}{a^{2}}{b^{2}}\CarlsonsymellintRF@{0}{c^{2}}{d^{2}}		 

int(0.5*int(1/(sqrt(t+(a*c + b*d)^(2))*sqrt(t+(a*d + b*c)^(2))*sqrt(t+4*a*b*c*d*(cosh(z))^(2))), t = 0..infinity), z = 0..infinity) = (1)/(2)*0.5*int(1/(sqrt(t+0)*sqrt(t+(a)^(2))*sqrt(t+(b)^(2))), t = 0..infinity)*0.5*int(1/(sqrt(t+0)*sqrt(t+(c)^(2))*sqrt(t+(d)^(2))), t = 0..infinity)

Integrate[EllipticF[ArcCos[Sqrt[(a*c + b*d)^(2)/4*a*b*c*d*(Cosh[z])^(2)]],(4*a*b*c*d*(Cosh[z])^(2)-(a*d + b*c)^(2))/(4*a*b*c*d*(Cosh[z])^(2)-(a*c + b*d)^(2))]/Sqrt[4*a*b*c*d*(Cosh[z])^(2)-(a*c + b*d)^(2)], {z, 0, Infinity}, GenerateConditions->None] == Divide[1,2]*EllipticF[ArcCos[Sqrt[0/(b)^(2)]],((b)^(2)-(a)^(2))/((b)^(2)-0)]/Sqrt[(b)^(2)-0]*EllipticF[ArcCos[Sqrt[0/(d)^(2)]],((d)^(2)-(c)^(2))/((d)^(2)-0)]/Sqrt[(d)^(2)-0]
Error Aborted - Skipped - Because timed out
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