19.7

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
19.7.E1	${\displaystyle{\displaystyle E\left(k\right){K^{\prime}}\left(k\right)+{E^{\prime}}\left(k\right)K\left(k\right)-K\left(k\right){K^{\prime}}\left(k\right)=\tfrac{1}{2}\pi}}$ \compellintEk@{k}\ccompellintKk@{k}+\ccompellintEk@{k}\compellintKk@{k}-\compellintKk@{k}\ccompellintKk@{k} = \tfrac{1}{2}\pi		EllipticE(k)*EllipticCK(k)+ EllipticCE(k)*EllipticK(k)- EllipticK(k)*EllipticCK(k) = (1)/(2)*Pi	EllipticE[(k)^2]*EllipticK[1-(k)^2]+ EllipticE[1-(k)^2]*EllipticK[(k)^2]- EllipticK[(k)^2]*EllipticK[1-(k)^2] == Divide[1,2]*Pi	Failure	Failure	Error	Failed [1 / 3] Result: Indeterminate Test Values: {Rule[k, 1]}
19.7#Ex1	${\displaystyle{\displaystyle K\left(ik/k^{\prime}\right)=k^{\prime}K\left(k\right)}}$ \compellintKk@{ik/k^{\prime}} = k^{\prime}\compellintKk@{k}		EllipticK(I*k/(sqrt(1 - (k)^(2)))) = sqrt(1 - (k)^(2))*EllipticK(k)	EllipticK[(I*k/(Sqrt[1 - (k)^(2)]))^2] == Sqrt[1 - (k)^(2)]*EllipticK[(k)^2]	Failure	Failure	Error	Failed [3 / 3] Result: Indeterminate Test Values: {Rule[k, 1]} Result: Complex[-2.220446049250313*^-16, -2.9198052634126777] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.7#Ex2	${\displaystyle{\displaystyle K\left(-ik^{\prime}/k\right)=kK\left(k^{\prime}\right)}}$ \compellintKk@{-ik^{\prime}/k} = k\compellintKk@{k^{\prime}}		EllipticK(- I*sqrt(1 - (k)^(2))/k) = k*EllipticK(sqrt(1 - (k)^(2)))	EllipticK[(- I*Sqrt[1 - (k)^(2)]/k)^2] == k*EllipticK[(Sqrt[1 - (k)^(2)])^2]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
19.7#Ex3	${\displaystyle{\displaystyle E\left(ik/k^{\prime}\right)=(1/k^{\prime})E\left(k\right)}}$ \compellintEk@{ik/k^{\prime}} = (1/k^{\prime})\compellintEk@{k}		EllipticE(I*k/(sqrt(1 - (k)^(2)))) = (1/(sqrt(1 - (k)^(2))))*EllipticE(k)	EllipticE[(I*k/(Sqrt[1 - (k)^(2)]))^2] == (1/(Sqrt[1 - (k)^(2)]))*EllipticE[(k)^2]	Failure	Failure	Failed [3 / 3] Result: Float(infinity)+Float(infinity)*I Test Values: {k = 1} Result: .6e-9+.4691535424*I Test Values: {k = 2} ... skip entries to safe data	Failed [3 / 3] Result: Indeterminate Test Values: {Rule[k, 1]} Result: Complex[-5.551115123125783*^-16, 0.46915354293820644] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.7#Ex4	${\displaystyle{\displaystyle E\left(-ik^{\prime}/k\right)=(1/k)E\left(k^{\prime}\right)}}$ \compellintEk@{-ik^{\prime}/k} = (1/k)\compellintEk@{k^{\prime}}		EllipticE(- I*sqrt(1 - (k)^(2))/k) = (1/k)*EllipticE(sqrt(1 - (k)^(2)))	EllipticE[(- I*Sqrt[1 - (k)^(2)]/k)^2] == (1/k)*EllipticE[(Sqrt[1 - (k)^(2)])^2]	Failure	Failure	Successful [Tested: 3]	Successful [Tested: 3]
19.7#Ex5	${\displaystyle{\displaystyle K\left(1/k\right)=k(K\left(k\right)-\mathrm{i}K\left(k^{\prime}\right))}}$ \compellintKk@{1/k} = k(\compellintKk@{k}-\iunit\compellintKk@{k^{\prime}})		EllipticK(1/k) = k*(EllipticK(k)- I*EllipticK(sqrt(1 - (k)^(2))))	EllipticK[(1/k)^2] == k*(EllipticK[(k)^2]- I*EllipticK[(Sqrt[1 - (k)^(2)])^2])	Failure	Failure	Error	Failed [3 / 3] Result: Indeterminate Test Values: {Rule[k, 1]} Result: Complex[-2.220446049250313*^-16, 4.313031294999287] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.7#Ex5	${\displaystyle{\displaystyle K\left(1/k\right)=k(K\left(k\right)+\mathrm{i}K\left(k^{\prime}\right))}}$ \compellintKk@{1/k} = k(\compellintKk@{k}+\iunit\compellintKk@{k^{\prime}})		EllipticK(1/k) = k*(EllipticK(k)+ I*EllipticK(sqrt(1 - (k)^(2))))	EllipticK[(1/k)^2] == k*(EllipticK[(k)^2]+ I*EllipticK[(Sqrt[1 - (k)^(2)])^2])	Failure	Failure	Error	Failed [1 / 3] Result: Indeterminate Test Values: {Rule[k, 1]}
19.7#Ex6	${\displaystyle{\displaystyle K\left(1/k^{\prime}\right)=k^{\prime}(K\left(k^{\prime}\right)+\mathrm{i}K\left(k\right))}}$ \compellintKk@{1/k^{\prime}} = k^{\prime}(\compellintKk@{k^{\prime}}+\iunit\compellintKk@{k})		EllipticK(1/(sqrt(1 - (k)^(2)))) = sqrt(1 - (k)^(2))*(EllipticK(sqrt(1 - (k)^(2)))+ I*EllipticK(k))	EllipticK[(1/(Sqrt[1 - (k)^(2)]))^2] == Sqrt[1 - (k)^(2)]*(EllipticK[(Sqrt[1 - (k)^(2)])^2]+ I*EllipticK[(k)^2])	Failure	Failure	Error	Failed [3 / 3] Result: Indeterminate Test Values: {Rule[k, 1]} Result: Complex[2.9198052634126785, -3.7351946687866775] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.7#Ex6	${\displaystyle{\displaystyle K\left(1/k^{\prime}\right)=k^{\prime}(K\left(k^{\prime}\right)-\mathrm{i}K\left(k\right))}}$ \compellintKk@{1/k^{\prime}} = k^{\prime}(\compellintKk@{k^{\prime}}-\iunit\compellintKk@{k})		EllipticK(1/(sqrt(1 - (k)^(2)))) = sqrt(1 - (k)^(2))*(EllipticK(sqrt(1 - (k)^(2)))- I*EllipticK(k))	EllipticK[(1/(Sqrt[1 - (k)^(2)]))^2] == Sqrt[1 - (k)^(2)]*(EllipticK[(Sqrt[1 - (k)^(2)])^2]- I*EllipticK[(k)^2])	Failure	Failure	Error	Failed [1 / 3] Result: Indeterminate Test Values: {Rule[k, 1]}
19.7#Ex7	${\displaystyle{\displaystyle E\left(1/k\right)=(1/k)\left(E\left(k\right)+\mathrm{i}E\left(k^{\prime}\right)-{k^{\prime}}^{2}K\left(k\right)-\mathrm{i}k^{2}K\left(k^{\prime}\right)\right)}}$ \compellintEk@{1/k} = (1/k)\left(\compellintEk@{k}+\iunit\compellintEk@{k^{\prime}}-{k^{\prime}}^{2}\compellintKk@{k}-\iunit k^{2}\compellintKk@{k^{\prime}}\right)		EllipticE(1/k) = (1/k)*(EllipticE(k)+ I*EllipticE(sqrt(1 - (k)^(2)))-1 - (k)^(2)*EllipticK(k)- I*(k)^(2)* EllipticK(sqrt(1 - (k)^(2))))	EllipticE[(1/k)^2] == (1/k)*(EllipticE[(k)^2]+ I*EllipticE[(Sqrt[1 - (k)^(2)])^2]-1 - (k)^(2)*EllipticK[(k)^2]- I*(k)^(2)* EllipticK[(Sqrt[1 - (k)^(2)])^2])	Failure	Failure	Error	Failed [3 / 3] Result: DirectedInfinity[] Test Values: {Rule[k, 1]} Result: Complex[3.4500631209220436, -1.8829831432620088] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.7#Ex7	${\displaystyle{\displaystyle E\left(1/k\right)=(1/k)\left(E\left(k\right)-\mathrm{i}E\left(k^{\prime}\right)-{k^{\prime}}^{2}K\left(k\right)+\mathrm{i}k^{2}K\left(k^{\prime}\right)\right)}}$ \compellintEk@{1/k} = (1/k)\left(\compellintEk@{k}-\iunit\compellintEk@{k^{\prime}}-{k^{\prime}}^{2}\compellintKk@{k}+\iunit k^{2}\compellintKk@{k^{\prime}}\right)		EllipticE(1/k) = (1/k)*(EllipticE(k)- I*EllipticE(sqrt(1 - (k)^(2)))-1 - (k)^(2)*EllipticK(k)+ I*(k)^(2)* EllipticK(sqrt(1 - (k)^(2))))	EllipticE[(1/k)^2] == (1/k)*(EllipticE[(k)^2]- I*EllipticE[(Sqrt[1 - (k)^(2)])^2]-1 - (k)^(2)*EllipticK[(k)^2]+ I*(k)^(2)* EllipticK[(Sqrt[1 - (k)^(2)])^2])	Failure	Failure	Error	Failed [3 / 3] Result: DirectedInfinity[] Test Values: {Rule[k, 1]} Result: Complex[3.4500631209220436, -3.773902383124376] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.7#Ex8	${\displaystyle{\displaystyle E\left(1/k^{\prime}\right)=(1/k^{\prime})\left(E\left(k^{\prime}\right)-\mathrm{i}E\left(k\right)-k^{2}K\left(k^{\prime}\right)+\mathrm{i}{k^{\prime}}^{2}K\left(k\right)\right)}}$ \compellintEk@{1/k^{\prime}} = (1/k^{\prime})\left(\compellintEk@{k^{\prime}}-\iunit\compellintEk@{k}-k^{2}\compellintKk@{k^{\prime}}+\iunit{k^{\prime}}^{2}\compellintKk@{k}\right)		EllipticE(1/(sqrt(1 - (k)^(2)))) = (1/(sqrt(1 - (k)^(2))))*(EllipticE(sqrt(1 - (k)^(2)))- I*EllipticE(k)- (k)^(2)* EllipticK(sqrt(1 - (k)^(2)))+ I*1 - (k)^(2)*EllipticK(k))	EllipticE[(1/(Sqrt[1 - (k)^(2)]))^2] == (1/(Sqrt[1 - (k)^(2)]))*(EllipticE[(Sqrt[1 - (k)^(2)])^2]- I*EllipticE[(k)^2]- (k)^(2)* EllipticK[(Sqrt[1 - (k)^(2)])^2]+ I*1 - (k)^(2)*EllipticK[(k)^2])	Failure	Failure	Error	Failed [3 / 3] Result: Indeterminate Test Values: {Rule[k, 1]} Result: Complex[-1.1384238737361991, -2.262384972182541] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.7#Ex8	${\displaystyle{\displaystyle E\left(1/k^{\prime}\right)=(1/k^{\prime})\left(E\left(k^{\prime}\right)+\mathrm{i}E\left(k\right)-k^{2}K\left(k^{\prime}\right)-\mathrm{i}{k^{\prime}}^{2}K\left(k\right)\right)}}$ \compellintEk@{1/k^{\prime}} = (1/k^{\prime})\left(\compellintEk@{k^{\prime}}+\iunit\compellintEk@{k}-k^{2}\compellintKk@{k^{\prime}}-\iunit{k^{\prime}}^{2}\compellintKk@{k}\right)		EllipticE(1/(sqrt(1 - (k)^(2)))) = (1/(sqrt(1 - (k)^(2))))*(EllipticE(sqrt(1 - (k)^(2)))+ I*EllipticE(k)- (k)^(2)* EllipticK(sqrt(1 - (k)^(2)))- I*1 - (k)^(2)*EllipticK(k))	EllipticE[(1/(Sqrt[1 - (k)^(2)]))^2] == (1/(Sqrt[1 - (k)^(2)]))*(EllipticE[(Sqrt[1 - (k)^(2)])^2]+ I*EllipticE[(k)^2]- (k)^(2)* EllipticK[(Sqrt[1 - (k)^(2)])^2]- I*1 - (k)^(2)*EllipticK[(k)^2])	Failure	Failure	Error	Failed [3 / 3] Result: Indeterminate Test Values: {Rule[k, 1]} Result: Complex[-0.45287687829515355, -3.814134176668458] Test Values: {Rule[k, 2]} ... skip entries to safe data
19.7#Ex19	${\displaystyle{\displaystyle F\left(i\phi,k\right)=iF\left(\psi,k^{\prime}\right)}}$ \incellintFk@{i\phi}{k} = i\incellintFk@{\psi}{k^{\prime}}		EllipticF(sin(I*phi), k) = I*EllipticF(sin(psi), sqrt(1 - (k)^(2)))	EllipticF[I*\[Phi], (k)^2] == I*EllipticF[\[Psi], (Sqrt[1 - (k)^(2)])^2]	Failure	Failure	Failed [300 / 300] Result: .1428695990-.263545696e-1*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, k = 1} Result: .749290340e-1-.334629029e-1*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[0.020142137049999537, -0.0010462389457662757] Test Values: {Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.015860617706546204, -0.003938067237051424] Test Values: {Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.7#Ex20	${\displaystyle{\displaystyle E\left(i\phi,k\right)=i\left(F\left(\psi,k^{\prime}\right)-E\left(\psi,k^{\prime}\right)+(\tan\psi)\sqrt{1-{k^{\prime}}^{2}{\sin^{2}}\psi}\right)}}$ \incellintEk@{i\phi}{k} = i\left(\incellintFk@{\psi}{k^{\prime}}-\incellintEk@{\psi}{k^{\prime}}+(\tan@@{\psi})\sqrt{1-{k^{\prime}}^{2}\sin^{2}@@{\psi}}\right)		EllipticE(sin(I*phi), k) = I*(EllipticF(sin(psi), sqrt(1 - (k)^(2)))- EllipticE(sin(psi), sqrt(1 - (k)^(2)))+(tan(psi))*sqrt(1 -1 - (k)^(2)*(sin(psi))^(2)))	EllipticE[I*\[Phi], (k)^2] == I*(EllipticF[\[Psi], (Sqrt[1 - (k)^(2)])^2]- EllipticE[\[Psi], (Sqrt[1 - (k)^(2)])^2]+(Tan[\[Psi]])*Sqrt[1 -1 - (k)^(2)*(Sin[\[Psi]])^(2)])	Failure	Failure	Failed [300 / 300] Result: -.9970133474-.1125517221*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, k = 1} Result: -2.257467281-.7782721018*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [300 / 300] Result: Complex[-0.3893501368763376, 0.20738614458301174] Test Values: {Rule[k, 1], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[-0.6710974690872284, 0.0060773305020283] Test Values: {Rule[k, 2], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data
19.7#Ex21	${\displaystyle{\displaystyle\Pi\left(i\phi,\alpha^{2},k\right)=i\left(F\left(\psi,k^{\prime}\right)-\alpha^{2}\Pi\left(\psi,1-\alpha^{2},k^{\prime}\right)\right)/{(1-\alpha^{2})}}}$ \incellintPik@{i\phi}{\alpha^{2}}{k} = i\left(\incellintFk@{\psi}{k^{\prime}}-\alpha^{2}\incellintPik@{\psi}{1-\alpha^{2}}{k^{\prime}}\right)/{(1-\alpha^{2})}		EllipticPi(sin(I*phi), (alpha)^(2), k) = I*(EllipticF(sin(psi), sqrt(1 - (k)^(2)))- (alpha)^(2)* EllipticPi(sin(psi), 1 - (alpha)^(2), sqrt(1 - (k)^(2))))/(1 - (alpha)^(2))	EllipticPi[\[Alpha]^(2), I*\[Phi],(k)^2] == I*(EllipticF[\[Psi], (Sqrt[1 - (k)^(2)])^2]- \[Alpha]^(2)* EllipticPi[1 - \[Alpha]^(2), \[Psi],(Sqrt[1 - (k)^(2)])^2])/(1 - \[Alpha]^(2))	Failure	Failure	Failed [292 / 300] Result: .926834363e-2-.484444094e-1*I Test Values: {alpha = 3/2, phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, k = 1} Result: -.130749569e-2-.277524276e-1*I Test Values: {alpha = 3/2, phi = 1/2*3^(1/2)+1/2*I, psi = 1/2*3^(1/2)+1/2*I, k = 2} ... skip entries to safe data	Failed [298 / 300] Result: Complex[0.013291772923717082, -0.006719909387202905] Test Values: {Rule[k, 1], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} Result: Complex[0.00953602334602252, -0.007394575555177196] Test Values: {Rule[k, 2], Rule[α, 1.5], Rule[ϕ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]], Rule[ψ, Times[Rational[1, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]]} ... skip entries to safe data