Elliptic Integrals - 19.18 Derivatives and Differential Equations

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
19.18.E1 ∂ ⁡ R F ⁡ ( x , y , z ) ∂ ⁡ z = - 1 6 ⁢ R D ⁡ ( x , y , z ) partial-derivative Carlson-integral-RF 𝑥 𝑦 𝑧 𝑧 1 6 Carlson-integral-RD 𝑥 𝑦 𝑧 {\displaystyle{\displaystyle\frac{\partial R_{F}\left(x,y,z\right)}{\partial z% }=-\tfrac{1}{6}R_{D}\left(x,y,z\right)}}
\pderiv{\CarlsonsymellintRF@{x}{y}{z}}{z} = -\tfrac{1}{6}\CarlsonsymellintRD@{x}{y}{z}		 

Error

(D[EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x], {temp, 1}]/.temp-> (x + y*I)) == -Divide[1,6]*3*(EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]-EllipticE[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)])/((x + y*I-y)*(x + y*I-x)^(1/2))
Missing Macro Error Failure -
Failed [18 / 18]
Result: Complex[0.03790163875178684, -0.07848225754688502]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}


Result: Complex[0.07302626282106058, 0.09607801553820669]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}

... skip entries to safe data
19.18.E2 d d x ⁡ R G ⁡ ( x + a , x + b , x + c ) = 1 2 ⁢ R F ⁡ ( x + a , x + b , x + c ) derivative 𝑥 Carlson-integral-RG 𝑥 𝑎 𝑥 𝑏 𝑥 𝑐 1 2 Carlson-integral-RF 𝑥 𝑎 𝑥 𝑏 𝑥 𝑐 {\displaystyle{\displaystyle\frac{\mathrm{d}}{\mathrm{d}x}R_{G}\left(x+a,x+b,x% +c\right)=\tfrac{1}{2}R_{F}\left(x+a,x+b,x+c\right)}}
\deriv{}{x}\CarlsonsymellintRG@{x+a}{x+b}{x+c} = \tfrac{1}{2}\CarlsonsymellintRF@{x+a}{x+b}{x+c}		 

Error

D[Sqrt[x + c-x + a]*(EllipticE[ArcCos[Sqrt[x + a/x + c]],(x + c-x + b)/(x + c-x + a)]+(Cot[ArcCos[Sqrt[x + a/x + c]]])^2*EllipticF[ArcCos[Sqrt[x + a/x + c]],(x + c-x + b)/(x + c-x + a)]+Cot[ArcCos[Sqrt[x + a/x + c]]]*Sqrt[1-k^2*Sin[ArcCos[Sqrt[x + a/x + c]]]^2]), x] == Divide[1,2]*EllipticF[ArcCos[Sqrt[x + a/x + c]],(x + c-x + b)/(x + c-x + a)]/Sqrt[x + c-x + a]
Missing Macro Error Aborted -
Failed [300 / 300]
Result: Plus[Complex[0.4534498410585545, 0.2544306388611797], Times[Complex[0.0, 1.7320508075688772], Plus[Complex[-0.5166444818917079, -0.6544984694978735], Times[Complex[0.0, 0.5892556509887895], Power[k, 2], Power[Plus[1.0, Times[-2.0, Power[k, 2]]], Rational[-1, 2]]], Times[Complex[0.0, -0.29462782549439476], Power[Plus[1.0, Times[-2.0, Power[k, 2]]], Rational[1, 2]]]]]]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[x, 1.5]}


Result: Plus[Complex[0.4534498410585545, 0.1389138883676965], Times[Complex[0.0, 1.7320508075688772], Plus[Complex[-1.7435577900831345, -0.43982297150257077], Times[Complex[0.0, 3.1304951684997055], Power[k, 2], Power[Plus[1.0, Times[-5.0, Power[k, 2]]], Rational[-1, 2]]], Times[Complex[0.0, -0.15652475842498526], Power[Plus[1.0, Times[-5.0, Power[k, 2]]], Rational[1, 2]]]]]]
Test Values: {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[x, 0.5]}

... skip entries to safe data
19.18.E9 ( x ⁢ ∂ ∂ ⁡ x + y ⁢ ∂ ∂ ⁡ y + z ⁢ ∂ ∂ ⁡ z ) ⁢ R F ⁡ ( x , y , z ) = - 1 2 ⁢ R F ⁡ ( x , y , z ) 𝑥 partial-derivative 𝑥 𝑦 partial-derivative 𝑦 𝑧 partial-derivative 𝑧 Carlson-integral-RF 𝑥 𝑦 𝑧 1 2 Carlson-integral-RF 𝑥 𝑦 𝑧 {\displaystyle{\displaystyle\left(x\frac{\partial}{\partial x}+y\frac{\partial% }{\partial y}+z\frac{\partial}{\partial z}\right)R_{F}\left(x,y,z\right)=-% \tfrac{1}{2}R_{F}\left(x,y,z\right)}}
\left(x\pderiv{}{x}+y\pderiv{}{y}+z\pderiv{}{z}\right)\CarlsonsymellintRF@{x}{y}{z} = -\tfrac{1}{2}\CarlsonsymellintRF@{x}{y}{z}		 

(x*diff(+ y*diff(+subs( temp=(x + y*I), diff( temp, temp$(1) ) ), y), x))*0.5*int(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+x + y*I)), t = 0..infinity) = -(1)/(2)*0.5*int(1/(sqrt(t+x)*sqrt(t+y)*sqrt(t+x + y*I)), t = 0..infinity)

(x*D[+ y*D[+(D[temp, {temp, 1}]/.temp-> (x + y*I)), y], x])*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x] == -Divide[1,2]*EllipticF[ArcCos[Sqrt[x/x + y*I]],(x + y*I-y)/(x + y*I-x)]/Sqrt[x + y*I-x]
Aborted Failure
Failed [18 / 18]
Result: -.8633499928+.6631327246*I
Test Values: {x = 3/2, y = -3/2}


Result: Float(infinity)+Float(infinity)*I
Test Values: {x = 3/2, y = 3/2}

... skip entries to safe data
Failed [18 / 18]
Result: Complex[-0.08107235486578032, 0.3392218839453487]
Test Values: {Rule[x, 1.5], Rule[y, -1.5]}


Result: Complex[-0.14411702330731, -0.3904606057684091]
Test Values: {Rule[x, 1.5], Rule[y, 1.5]}

... skip entries to safe data
19.18.E14 ∂ 2 ⁡ w ∂ ⁡ x 2 = ∂ 2 ⁡ w ∂ ⁡ y 2 + 1 y ⁢ ∂ ⁡ w ∂ ⁡ y partial-derivative 𝑤 𝑥 2 partial-derivative 𝑤 𝑦 2 1 𝑦 partial-derivative 𝑤 𝑦 {\displaystyle{\displaystyle\frac{{\partial}^{2}w}{{\partial x}^{2}}=\frac{{% \partial}^{2}w}{{\partial y}^{2}}+\frac{1}{y}\frac{\partial w}{\partial y}}}
\pderiv[2]{w}{x} = \pderiv[2]{w}{y}+\frac{1}{y}\pderiv{w}{y}		 

diff(w, [x$(2)]) = diff(w, [y$(2)])+(1)/(y)*diff(w, y)

D[w, {x, 2}] == D[w, {y, 2}]+Divide[1,y]*D[w, y]
Successful Successful - Successful [Tested: 180]
19.18.E15 ∂ 2 ⁡ W ∂ ⁡ t 2 = ∂ 2 ⁡ W ∂ ⁡ x 2 + ∂ 2 ⁡ W ∂ ⁡ y 2 partial-derivative 𝑊 𝑡 2 partial-derivative 𝑊 𝑥 2 partial-derivative 𝑊 𝑦 2 {\displaystyle{\displaystyle\frac{{\partial}^{2}W}{{\partial t}^{2}}=\frac{{% \partial}^{2}W}{{\partial x}^{2}}+\frac{{\partial}^{2}W}{{\partial y}^{2}}}}
\pderiv[2]{W}{t} = \pderiv[2]{W}{x}+\pderiv[2]{W}{y}		 

diff(W, [t$(2)]) = diff(W, [x$(2)])+ diff(W, [y$(2)])

D[W, {t, 2}] == D[W, {x, 2}]+ D[W, {y, 2}]
Successful Successful - Successful [Tested: 300]
19.18.E16 ∂ 2 ⁡ u ∂ ⁡ x 2 + ∂ 2 ⁡ u ∂ ⁡ y 2 + 1 y ⁢ ∂ ⁡ u ∂ ⁡ y = 0 partial-derivative 𝑢 𝑥 2 partial-derivative 𝑢 𝑦 2 1 𝑦 partial-derivative 𝑢 𝑦 0 {\displaystyle{\displaystyle\frac{{\partial}^{2}u}{{\partial x}^{2}}+\frac{{% \partial}^{2}u}{{\partial y}^{2}}+\frac{1}{y}\frac{\partial u}{\partial y}=0}}
\pderiv[2]{u}{x}+\pderiv[2]{u}{y}+\frac{1}{y}\pderiv{u}{y} = 0		 

diff(u, [x$(2)])+ diff(u, [y$(2)])+(1)/(y)*diff(u, y) = 0

D[u, {x, 2}]+ D[u, {y, 2}]+Divide[1,y]*D[u, y] == 0
Successful Successful - Successful [Tested: 180]
19.18.E17 ∂ 2 ⁡ U ∂ ⁡ x 2 + ∂ 2 ⁡ U ∂ ⁡ y 2 + ∂ 2 ⁡ U ∂ ⁡ z 2 = 0 partial-derivative 𝑈 𝑥 2 partial-derivative 𝑈 𝑦 2 partial-derivative 𝑈 𝑧 2 0 {\displaystyle{\displaystyle\frac{{\partial}^{2}U}{{\partial x}^{2}}+\frac{{% \partial}^{2}U}{{\partial y}^{2}}+\frac{{\partial}^{2}U}{{\partial z}^{2}}=0}}
\pderiv[2]{U}{x}+\pderiv[2]{U}{y}+\pderiv[2]{U}{z} = 0		 

diff(U, [x$(2)])+ diff(U, [y$(2)])+ subs( temp=(x + y*I), diff( U, temp$(2) ) ) = 0

D[U, {x, 2}]+ D[U, {y, 2}]+ (D[U, {temp, 2}]/.temp-> (x + y*I)) == 0
Successful Successful - Successful [Tested: 180]
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