19.4

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
19.4#Ex1	${\displaystyle{\displaystyle\frac{\mathrm{d}K\left(k\right)}{\mathrm{d}k}=\frac{E\left(k\right)-{k^{\prime}}^{2}K\left(k\right)}{k{k^{\prime}}^{2}}}}$ \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	${\displaystyle{\displaystyle\frac{\mathrm{d}(E\left(k\right)-{k^{\prime}}^{2}K\left(k\right))}{\mathrm{d}k}=kK\left(k\right)}}$ \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	${\displaystyle{\displaystyle\frac{\mathrm{d}E\left(k\right)}{\mathrm{d}k}=\frac{E\left(k\right)-K\left(k\right)}{k}}}$ \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	${\displaystyle{\displaystyle\frac{\mathrm{d}(E\left(k\right)-K\left(k\right))}{\mathrm{d}k}=-\frac{kE\left(k\right)}{{k^{\prime}}^{2}}}}$ \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	${\displaystyle{\displaystyle\frac{{\mathrm{d}}^{2}E\left(k\right)}{{\mathrm{d}k}^{2}}=-\frac{1}{k}\frac{\mathrm{d}K\left(k\right)}{\mathrm{d}k}}}$ \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	${\displaystyle{\displaystyle-\frac{1}{k}\frac{\mathrm{d}K\left(k\right)}{\mathrm{d}k}=\frac{{k^{\prime}}^{2}K\left(k\right)-E\left(k\right)}{k^{2}{k^{\prime}}^{2}}}}$ -\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	${\displaystyle{\displaystyle\frac{\partial\Pi\left(\alpha^{2},k\right)}{\partial k}=\frac{k}{{k^{\prime}}^{2}(k^{2}-\alpha^{2})}(E\left(k\right)-{k^{\prime}}^{2}\Pi\left(\alpha^{2},k\right))}}$ \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	${\displaystyle{\displaystyle\frac{\partial F\left(\phi,k\right)}{\partial k}={\frac{E\left(\phi,k\right)-{k^{\prime}}^{2}F\left(\phi,k\right)}{k{k^{\prime}}^{2}}-\frac{k\sin\phi\cos\phi}{{k^{\prime}}^{2}\sqrt{1-k^{2}{\sin^{2}}\phi}}}}}$ \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	${\displaystyle{\displaystyle\frac{\partial E\left(\phi,k\right)}{\partial k}=\frac{E\left(\phi,k\right)-F\left(\phi,k\right)}{k}}}$ \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	${\displaystyle{\displaystyle\frac{\partial\Pi\left(\phi,\alpha^{2},k\right)}{\partial k}=\frac{k}{{k^{\prime}}^{2}(k^{2}-\alpha^{2})}\left({E\left(\phi,k\right)-{k^{\prime}}^{2}\Pi\left(\phi,\alpha^{2},k\right)}-\frac{k^{2}\sin\phi\cos\phi}{\sqrt{1-k^{2}{\sin^{2}}\phi}}\right)}}$ \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	${\displaystyle{\displaystyle(k{k^{\prime}}^{2}D_{k}^{2}+(1-3k^{2})D_{k}-k)F\left(\phi,k\right)=\frac{-k\sin\phi\cos\phi}{(1-k^{2}{\sin^{2}}\phi)^{3/2}}}}$ (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	${\displaystyle{\displaystyle(k{k^{\prime}}^{2}D_{k}^{2}+{k^{\prime}}^{2}D_{k}+k)E\left(\phi,k\right)=\frac{k\sin\phi\cos\phi}{\sqrt{1-k^{2}{\sin^{2}}\phi}}}}$ (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