Elliptic Integrals - 19.4 Derivatives and Differential Equations

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
19.4#Ex1 d K ⁡ ( k ) d k = E ⁡ ( k ) - k ′ 2 ⁢ K ⁡ ( k ) k ⁢ k ′ 2 derivative complete-elliptic-integral-first-kind-K 𝑘 𝑘 complete-elliptic-integral-second-kind-E 𝑘 superscript superscript 𝑘 ′ 2 complete-elliptic-integral-first-kind-K 𝑘 𝑘 superscript superscript 𝑘 ′ 2 {\displaystyle{\displaystyle\frac{\mathrm{d}K\left(k\right)}{\mathrm{d}k}=% \frac{E\left(k\right)-{k^{\prime}}^{2}K\left(k\right)}{k{k^{\prime}}^{2}}}}
\deriv{\compellintKk@{k}}{k} = \frac{\compellintEk@{k}-{k^{\prime}}^{2}\compellintKk@{k}}{k{k^{\prime}}^{2}}		 

diff(EllipticK(k), k) = (EllipticE(k)-1 - (k)^(2)*EllipticK(k))/(k*1 - (k)^(2))

D[EllipticK[(k)^2], k] == Divide[EllipticE[(k)^2]-1 - (k)^(2)*EllipticK[(k)^2],k*1 - (k)^(2)]
Failure Failure Error
Failed [3 / 3]
Result: Indeterminate
Test Values: {Rule[k, 1]}


Result: Complex[-2.4717549813624253, 3.1435959698369205]
Test Values: {Rule[k, 2]}

... skip entries to safe data
19.4#Ex2 d ( E ⁡ ( k ) - k ′ 2 ⁢ K ⁡ ( k ) ) d k = k ⁢ K ⁡ ( k ) derivative complete-elliptic-integral-second-kind-E 𝑘 superscript superscript 𝑘 ′ 2 complete-elliptic-integral-first-kind-K 𝑘 𝑘 𝑘 complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle\frac{\mathrm{d}(E\left(k\right)-{k^{\prime}}^{2}K% \left(k\right))}{\mathrm{d}k}=kK\left(k\right)}}
\deriv{(\compellintEk@{k}-{k^{\prime}}^{2}\compellintKk@{k})}{k} = k\compellintKk@{k}		 

diff(EllipticE(k)-1 - (k)^(2)*EllipticK(k), k) = k*EllipticK(k)

D[EllipticE[(k)^2]-1 - (k)^(2)*EllipticK[(k)^2], k] == k*EllipticK[(k)^2]
Failure Failure Error
Failed [3 / 3]
Result: Indeterminate
Test Values: {Rule[k, 1]}


Result: Complex[-3.3189229307917216, 6.419990143492479]
Test Values: {Rule[k, 2]}

... skip entries to safe data
19.4#Ex3 d E ⁡ ( k ) d k = E ⁡ ( k ) - K ⁡ ( k ) k derivative complete-elliptic-integral-second-kind-E 𝑘 𝑘 complete-elliptic-integral-second-kind-E 𝑘 complete-elliptic-integral-first-kind-K 𝑘 𝑘 {\displaystyle{\displaystyle\frac{\mathrm{d}E\left(k\right)}{\mathrm{d}k}=% \frac{E\left(k\right)-K\left(k\right)}{k}}}
\deriv{\compellintEk@{k}}{k} = \frac{\compellintEk@{k}-\compellintKk@{k}}{k}		 

diff(EllipticE(k), k) = (EllipticE(k)- EllipticK(k))/(k)

D[EllipticE[(k)^2], k] == Divide[EllipticE[(k)^2]- EllipticK[(k)^2],k]
Successful Successful - Successful [Tested: 3]
19.4#Ex4 d ( E ⁡ ( k ) - K ⁡ ( k ) ) d k = - k ⁢ E ⁡ ( k ) k ′ 2 derivative complete-elliptic-integral-second-kind-E 𝑘 complete-elliptic-integral-first-kind-K 𝑘 𝑘 𝑘 complete-elliptic-integral-second-kind-E 𝑘 superscript superscript 𝑘 ′ 2 {\displaystyle{\displaystyle\frac{\mathrm{d}(E\left(k\right)-K\left(k\right))}% {\mathrm{d}k}=-\frac{kE\left(k\right)}{{k^{\prime}}^{2}}}}
\deriv{(\compellintEk@{k}-\compellintKk@{k})}{k} = -\frac{k\compellintEk@{k}}{{k^{\prime}}^{2}}		 

diff(EllipticE(k)- EllipticK(k), k) = -(k*EllipticE(k))/(1 - (k)^(2))

D[EllipticE[(k)^2]- EllipticK[(k)^2], k] == -Divide[k*EllipticE[(k)^2],1 - (k)^(2)]
Successful Successful -
Failed [1 / 3]
Result: Indeterminate
Test Values: {Rule[k, 1]}

19.4.E3 d 2 E ⁡ ( k ) d k 2 = - 1 k ⁢ d K ⁡ ( k ) d k derivative complete-elliptic-integral-second-kind-E 𝑘 𝑘 2 1 𝑘 derivative complete-elliptic-integral-first-kind-K 𝑘 𝑘 {\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}E\left(k\right)}{{\mathrm{d}% k}^{2}}=-\frac{1}{k}\frac{\mathrm{d}K\left(k\right)}{\mathrm{d}k}}}
\deriv[2]{\compellintEk@{k}}{k} = -\frac{1}{k}\deriv{\compellintKk@{k}}{k}		 

diff(EllipticE(k), [k$(2)]) = -(1)/(k)*diff(EllipticK(k), k)

D[EllipticE[(k)^2], {k, 2}] == -Divide[1,k]*D[EllipticK[(k)^2], k]
Successful Successful -
Failed [1 / 3]
Result: Indeterminate
Test Values: {Rule[k, 1]}

19.4.E3 - 1 k ⁢ d K ⁡ ( k ) d k = k ′ 2 ⁢ K ⁡ ( k ) - E ⁡ ( k ) k 2 ⁢ k ′ 2 1 𝑘 derivative complete-elliptic-integral-first-kind-K 𝑘 𝑘 superscript superscript 𝑘 ′ 2 complete-elliptic-integral-first-kind-K 𝑘 complete-elliptic-integral-second-kind-E 𝑘 superscript 𝑘 2 superscript superscript 𝑘 ′ 2 {\displaystyle{\displaystyle-\frac{1}{k}\frac{\mathrm{d}K\left(k\right)}{% \mathrm{d}k}=\frac{{k^{\prime}}^{2}K\left(k\right)-E\left(k\right)}{k^{2}{k^{% \prime}}^{2}}}}
-\frac{1}{k}\deriv{\compellintKk@{k}}{k} = \frac{{k^{\prime}}^{2}\compellintKk@{k}-\compellintEk@{k}}{k^{2}{k^{\prime}}^{2}}		 

-(1)/(k)*diff(EllipticK(k), k) = (1 - (k)^(2)*EllipticK(k)- EllipticE(k))/((k)^(2)*1 - (k)^(2))

-Divide[1,k]*D[EllipticK[(k)^2], k] == Divide[1 - (k)^(2)*EllipticK[(k)^2]- EllipticE[(k)^2],(k)^(2)*1 - (k)^(2)]
Error Failure -
Failed [3 / 3]
Result: DirectedInfinity[]
Test Values: {Rule[k, 1]}


Result: DirectedInfinity[]
Test Values: {Rule[k, 2]}

... skip entries to safe data
19.4.E4 ∂ ⁡ Π ⁡ ( α 2 , k ) ∂ ⁡ k = k k ′ 2 ⁢ ( k 2 - α 2 ) ⁢ ( E ⁡ ( k ) - k ′ 2 ⁢ Π ⁡ ( α 2 , k ) ) partial-derivative complete-elliptic-integral-third-kind-Pi superscript 𝛼 2 𝑘 𝑘 𝑘 superscript superscript 𝑘 ′ 2 superscript 𝑘 2 superscript 𝛼 2 complete-elliptic-integral-second-kind-E 𝑘 superscript superscript 𝑘 ′ 2 complete-elliptic-integral-third-kind-Pi superscript 𝛼 2 𝑘 {\displaystyle{\displaystyle\frac{\partial\Pi\left(\alpha^{2},k\right)}{% \partial k}=\frac{k}{{k^{\prime}}^{2}(k^{2}-\alpha^{2})}(E\left(k\right)-{k^{% \prime}}^{2}\Pi\left(\alpha^{2},k\right))}}
\pderiv{\compellintPik@{\alpha^{2}}{k}}{k} = \frac{k}{{k^{\prime}}^{2}(k^{2}-\alpha^{2})}(\compellintEk@{k}-{k^{\prime}}^{2}\compellintPik@{\alpha^{2}}{k})		 

diff(EllipticPi((alpha)^(2), k), k) = (k)/(1 - (k)^(2)*((k)^(2)- (alpha)^(2)))*(EllipticE(k)-1 - (k)^(2)*EllipticPi((alpha)^(2), k))

D[EllipticPi[\[Alpha]^(2), (k)^2], k] == Divide[k,1 - (k)^(2)*((k)^(2)- \[Alpha]^(2))]*(EllipticE[(k)^2]-1 - (k)^(2)*EllipticPi[\[Alpha]^(2), (k)^2])
Failure Failure Error
Failed [9 / 9]
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[α, 1.5]}


Result: Complex[0.38994760629924174, 1.2322724929931343]
Test Values: {Rule[k, 2], Rule[α, 1.5]}

... skip entries to safe data
19.4.E5 ∂ ⁡ F ⁡ ( ϕ , k ) ∂ ⁡ k = E ⁡ ( ϕ , k ) - k ′ 2 ⁢ F ⁡ ( ϕ , k ) k ⁢ k ′ 2 - k ⁢ sin ⁡ ϕ ⁢ cos ⁡ ϕ k ′ 2 ⁢ 1 - k 2 ⁢ sin 2 ⁡ ϕ partial-derivative elliptic-integral-first-kind-F italic-ϕ 𝑘 𝑘 elliptic-integral-second-kind-E italic-ϕ 𝑘 superscript superscript 𝑘 ′ 2 elliptic-integral-first-kind-F italic-ϕ 𝑘 𝑘 superscript superscript 𝑘 ′ 2 𝑘 italic-ϕ italic-ϕ superscript superscript 𝑘 ′ 2 1 superscript 𝑘 2 2 italic-ϕ {\displaystyle{\displaystyle\frac{\partial F\left(\phi,k\right)}{\partial k}={% \frac{E\left(\phi,k\right)-{k^{\prime}}^{2}F\left(\phi,k\right)}{k{k^{\prime}}% ^{2}}-\frac{k\sin\phi\cos\phi}{{k^{\prime}}^{2}\sqrt{1-k^{2}{\sin^{2}}\phi}}}}}
\pderiv{\incellintFk@{\phi}{k}}{k} = {\frac{\incellintEk@{\phi}{k}-{k^{\prime}}^{2}\incellintFk@{\phi}{k}}{k{k^{\prime}}^{2}}-\frac{k\sin@@{\phi}\cos@@{\phi}}{{k^{\prime}}^{2}\sqrt{1-k^{2}\sin^{2}@@{\phi}}}}		 

diff(EllipticF(sin(phi), k), k) = (EllipticE(sin(phi), k)-1 - (k)^(2)*EllipticF(sin(phi), k))/(k*1 - (k)^(2))-(k*sin(phi)*cos(phi))/(1 - (k)^(2)*sqrt(1 - (k)^(2)* (sin(phi))^(2)))

D[EllipticF[\[Phi], (k)^2], k] == Divide[EllipticE[\[Phi], (k)^2]-1 - (k)^(2)*EllipticF[\[Phi], (k)^2],k*1 - (k)^(2)]-Divide[k*Sin[\[Phi]]*Cos[\[Phi]],1 - (k)^(2)*Sqrt[1 - (k)^(2)* (Sin[\[Phi]])^(2)]]
Failure Failure
Failed [30 / 30]
Result: Float(infinity)+Float(infinity)*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, k = 1}


Result: -1.296981010-1.781988683*I
Test Values: {phi = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [30 / 30]
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[-1.2927667883728842, -0.7915995039632082]
Test Values: {Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.4.E6 ∂ ⁡ E ⁡ ( ϕ , k ) ∂ ⁡ k = E ⁡ ( ϕ , k ) - F ⁡ ( ϕ , k ) k partial-derivative elliptic-integral-second-kind-E italic-ϕ 𝑘 𝑘 elliptic-integral-second-kind-E italic-ϕ 𝑘 elliptic-integral-first-kind-F italic-ϕ 𝑘 𝑘 {\displaystyle{\displaystyle\frac{\partial E\left(\phi,k\right)}{\partial k}=% \frac{E\left(\phi,k\right)-F\left(\phi,k\right)}{k}}}
\pderiv{\incellintEk@{\phi}{k}}{k} = \frac{\incellintEk@{\phi}{k}-\incellintFk@{\phi}{k}}{k}		 

diff(EllipticE(sin(phi), k), k) = (EllipticE(sin(phi), k)- EllipticF(sin(phi), k))/(k)

D[EllipticE[\[Phi], (k)^2], k] == Divide[EllipticE[\[Phi], (k)^2]- EllipticF[\[Phi], (k)^2],k]
Successful Successful - Successful [Tested: 30]
19.4.E7 ∂ ⁡ Π ⁡ ( ϕ , α 2 , k ) ∂ ⁡ k = k k ′ 2 ⁢ ( k 2 - α 2 ) ⁢ ( E ⁡ ( ϕ , k ) - k ′ 2 ⁢ Π ⁡ ( ϕ , α 2 , k ) - k 2 ⁢ sin ⁡ ϕ ⁢ cos ⁡ ϕ 1 - k 2 ⁢ sin 2 ⁡ ϕ ) partial-derivative elliptic-integral-third-kind-Pi italic-ϕ superscript 𝛼 2 𝑘 𝑘 𝑘 superscript superscript 𝑘 ′ 2 superscript 𝑘 2 superscript 𝛼 2 elliptic-integral-second-kind-E italic-ϕ 𝑘 superscript superscript 𝑘 ′ 2 elliptic-integral-third-kind-Pi italic-ϕ superscript 𝛼 2 𝑘 superscript 𝑘 2 italic-ϕ italic-ϕ 1 superscript 𝑘 2 2 italic-ϕ {\displaystyle{\displaystyle\frac{\partial\Pi\left(\phi,\alpha^{2},k\right)}{% \partial k}=\frac{k}{{k^{\prime}}^{2}(k^{2}-\alpha^{2})}\left({E\left(\phi,k% \right)-{k^{\prime}}^{2}\Pi\left(\phi,\alpha^{2},k\right)}-\frac{k^{2}\sin\phi% \cos\phi}{\sqrt{1-k^{2}{\sin^{2}}\phi}}\right)}}
\pderiv{\incellintPik@{\phi}{\alpha^{2}}{k}}{k} = \frac{k}{{k^{\prime}}^{2}(k^{2}-\alpha^{2})}\left({\incellintEk@{\phi}{k}-{k^{\prime}}^{2}\incellintPik@{\phi}{\alpha^{2}}{k}}-\frac{k^{2}\sin@@{\phi}\cos@@{\phi}}{\sqrt{1-k^{2}\sin^{2}@@{\phi}}}\right)		 

diff(EllipticPi(sin(phi), (alpha)^(2), k), k) = (k)/(1 - (k)^(2)*((k)^(2)- (alpha)^(2)))*(EllipticE(sin(phi), k)-1 - (k)^(2)*EllipticPi(sin(phi), (alpha)^(2), k)-((k)^(2)* sin(phi)*cos(phi))/(sqrt(1 - (k)^(2)* (sin(phi))^(2))))

D[EllipticPi[\[Alpha]^(2), \[Phi],(k)^2], k] == Divide[k,1 - (k)^(2)*((k)^(2)- \[Alpha]^(2))]*(EllipticE[\[Phi], (k)^2]-1 - (k)^(2)*EllipticPi[\[Alpha]^(2), \[Phi],(k)^2]-Divide[(k)^(2)* Sin[\[Phi]]*Cos[\[Phi]],Sqrt[1 - (k)^(2)* (Sin[\[Phi]])^(2)]])
Failure Aborted
Failed [90 / 90]
Result: Float(undefined)+Float(undefined)*I
Test Values: {alpha = 3/2, phi = 1/2*3^(1/2)+1/2*I, k = 1}


Result: -5.135398794+1.052011331*I
Test Values: {alpha = 3/2, phi = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [90 / 90]
Result: Indeterminate
Test Values: {Rule[k, 1], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Complex[-1.1264235284707635, -0.9763567309038728]
Test Values: {Rule[k, 2], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.4.E8 ( k ⁢ k ′ 2 ⁢ D k 2 + ( 1 - 3 ⁢ k 2 ) ⁢ D k - k ) ⁢ F ⁡ ( ϕ , k ) = - k ⁢ sin ⁡ ϕ ⁢ cos ⁡ ϕ ( 1 - k 2 ⁢ sin 2 ⁡ ϕ ) 3 / 2 𝑘 superscript superscript 𝑘 ′ 2 superscript subscript 𝐷 𝑘 2 1 3 superscript 𝑘 2 subscript 𝐷 𝑘 𝑘 elliptic-integral-first-kind-F italic-ϕ 𝑘 𝑘 italic-ϕ italic-ϕ superscript 1 superscript 𝑘 2 2 italic-ϕ 3 2 {\displaystyle{\displaystyle(k{k^{\prime}}^{2}D_{k}^{2}+(1-3k^{2})D_{k}-k)F% \left(\phi,k\right)=\frac{-k\sin\phi\cos\phi}{(1-k^{2}{\sin^{2}}\phi)^{3/2}}}}
(k{k^{\prime}}^{2}D_{k}^{2}+(1-3k^{2})D_{k}-k)\incellintFk@{\phi}{k} = \frac{-k\sin@@{\phi}\cos@@{\phi}}{(1-k^{2}\sin^{2}@@{\phi})^{3/2}}		 

(k*1 - (k)^(2)*(D[k])^(2)+(1 - 3*(k)^(2))*D[k]- k)*EllipticF(sin(phi), k) = (- k*sin(phi)*cos(phi))/((1 - (k)^(2)* (sin(phi))^(2))^(3/2))

(k*1 - (k)^(2)*(Subscript[D, k])^(2)+(1 - 3*(k)^(2))*Subscript[D, k]- k)*EllipticF[\[Phi], (k)^2] == Divide[- k*Sin[\[Phi]]*Cos[\[Phi]],(1 - (k)^(2)* (Sin[\[Phi]])^(2))^(3/2)]
Error Failure -
Failed [300 / 300]
Result: Plus[Complex[0.4174282354972822, 0.36074991075375373], Times[Complex[0.43180375739814203, 0.27142936483528934], Plus[Complex[-0.8660254037844387, -0.49999999999999994], Times[Complex[-0.12500000000000003, -0.21650635094610965], D]]]]
Test Values: {Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[D, k], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Plus[Complex[-0.38000132033999284, 0.977947559972491], Times[Complex[0.3965687056216178, 0.33175091278780894], Plus[Complex[-4.763139720814413, -2.7499999999999996], Times[Complex[-0.5000000000000001, -0.8660254037844386], D]]]]
Test Values: {Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[D, k], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
19.4.E9 ( k ⁢ k ′ 2 ⁢ D k 2 + k ′ 2 ⁢ D k + k ) ⁢ E ⁡ ( ϕ , k ) = k ⁢ sin ⁡ ϕ ⁢ cos ⁡ ϕ 1 - k 2 ⁢ sin 2 ⁡ ϕ 𝑘 superscript superscript 𝑘 ′ 2 superscript subscript 𝐷 𝑘 2 superscript superscript 𝑘 ′ 2 subscript 𝐷 𝑘 𝑘 elliptic-integral-second-kind-E italic-ϕ 𝑘 𝑘 italic-ϕ italic-ϕ 1 superscript 𝑘 2 2 italic-ϕ {\displaystyle{\displaystyle(k{k^{\prime}}^{2}D_{k}^{2}+{k^{\prime}}^{2}D_{k}+% k)E\left(\phi,k\right)=\frac{k\sin\phi\cos\phi}{\sqrt{1-k^{2}{\sin^{2}}\phi}}}}
(k{k^{\prime}}^{2}D_{k}^{2}+{k^{\prime}}^{2}D_{k}+k)\incellintEk@{\phi}{k} = \frac{k\sin@@{\phi}\cos@@{\phi}}{\sqrt{1-k^{2}\sin^{2}@@{\phi}}}		 

(k*1 - (k)^(2)*(D[k])^(2)+1 - (k)^(2)*D[k]+ k)*EllipticE(sin(phi), k) = (k*sin(phi)*cos(phi))/(sqrt(1 - (k)^(2)* (sin(phi))^(2)))

(k*1 - (k)^(2)*(Subscript[D, k])^(2)+1 - (k)^(2)*Subscript[D, k]+ k)*EllipticE[\[Phi], (k)^2] == Divide[k*Sin[\[Phi]]*Cos[\[Phi]],Sqrt[1 - (k)^(2)* (Sin[\[Phi]])^(2)]]
Error Failure -
Failed [300 / 300]
Result: Plus[Complex[-0.4327885168580316, -0.2292976446734403], Times[Complex[0.43278851685803155, 0.22929764467344024], Plus[Complex[2.566987298107781, -0.24999999999999997], Times[Complex[-0.12500000000000003, -0.21650635094610965], D]]]]
Test Values: {Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[D, k], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}


Result: Plus[Complex[-0.6011783848834926, -0.7526006723022071], Times[Complex[0.44208095936294645, 0.16535187593702125], Plus[Complex[3.2679491924311224, -0.9999999999999999], Times[Complex[-0.5000000000000001, -0.8660254037844386], D]]]]
Test Values: {Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[Subscript[D, k], Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]}

... skip entries to safe data
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