22.16

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
22.16.E1	${\displaystyle{\displaystyle\operatorname{am}\left(x,k\right)=\operatorname{Arcsin}\left(\operatorname{sn}\left(x,k\right)\right)}}$ \Jacobiamk@{x}{k} = \Asin@{\Jacobiellsnk@{x}{k}}		Error	JacobiAmplitude[x, Power[k, 2]] == ArcSin[JacobiSN[x, (k)^2]]	Missing Macro Error	Failure	-	Failed [1 / 3] Result: 6.283185307179586 Test Values: {Rule[k, 3], Rule[x, Rational[3, 2]]}
22.16.E2	${\displaystyle{\displaystyle\operatorname{am}\left(x+2K,k\right)=\operatorname{am}\left(x,k\right)+\pi}}$ \Jacobiamk@{x+2K}{k} = \Jacobiamk@{x}{k}+\pi		JacobiAM(x + 2*EllipticK(k), k) = JacobiAM(x, k)+ Pi	JacobiAmplitude[x + 2*EllipticK[(k)^2], Power[k, 2]] == JacobiAmplitude[x, Power[k, 2]]+ Pi	Failure	Failure	Error	Failed [1 / 3] Result: Plus[-4.273320998840302, Gudermannian[DirectedInfinity[]]] Test Values: {Rule[k, 1], Rule[x, Rational[3, 2]]}
22.16.E3	${\displaystyle{\displaystyle\operatorname{am}\left(x,k\right)=\int_{0}^{x}\operatorname{dn}\left(t,k\right)\mathrm{d}t}}$ \Jacobiamk@{x}{k} = \int_{0}^{x}\Jacobielldnk@{t}{k}\diff{t}		JacobiAM(x, k) = int(JacobiDN(t, k), t = 0..x)	JacobiAmplitude[x, Power[k, 2]] == Integrate[JacobiDN[t, (k)^2], {t, 0, x}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 9]	Successful [Tested: 9]
22.16.E4	${\displaystyle{\displaystyle\operatorname{am}\left(x,0\right)=x}}$ \Jacobiamk@{x}{0} = x		JacobiAM(x, 0) = x	JacobiAmplitude[x, Power[0, 2]] == x	Successful	Successful	-	Successful [Tested: 3]
22.16.E5	${\displaystyle{\displaystyle\operatorname{am}\left(x,1\right)=\operatorname{gd}\left(x\right)}}$ \Jacobiamk@{x}{1} = \Gudermannian@{x}	${\displaystyle{\displaystyle-\infty<x,x<\infty}}$	JacobiAM(x, 1) = arctan(sinh(x))	JacobiAmplitude[x, Power[1, 2]] == Gudermannian[x]	Failure	Successful	Successful [Tested: 3]	Successful [Tested: 3]
22.16.E9	${\displaystyle{\displaystyle\operatorname{am}\left(x,k\right)=\frac{\pi}{2K}x+2\sum_{n=1}^{\infty}\frac{q^{n}\sin\left(2n\zeta\right)}{n(1+q^{2n})}}}$ \Jacobiamk@{x}{k} = \frac{\pi}{2K}x+2\sum_{n=1}^{\infty}\frac{q^{n}\sin@{2n\zeta}}{n(1+q^{2n})}		JacobiAM(x, k) = (Pi)/(2*EllipticK(k))*x + 2*sum(((exp(- Pi*EllipticCK(k)/EllipticK(k)))^(n)* sin(2*n*zeta))/(n*(1 +(exp(- Pi*EllipticCK(k)/EllipticK(k)))^(2*n))), n = 1..infinity)	JacobiAmplitude[x, Power[k, 2]] == Divide[Pi,2*EllipticK[(k)^2]]*x + 2*Sum[Divide[(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(n)* Sin[2*n*\[Zeta]],n*(1 +(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])^(2*n))], {n, 1, Infinity}, GenerateConditions->None]	Failure	Failure	Skipped - Because timed out	Failed [30 / 30] Result: Plus[Complex[1.9977537490349477, 0.49999999999999994], Times[-1.0, Gudermannian[DirectedInfinity[]]]] Test Values: {Rule[k, 1], Rule[x, Rational[3, 2]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.6288351638274511, -0.8359897636003678] Test Values: {Rule[k, 2], Rule[x, Rational[3, 2]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
22.16.E10	${\displaystyle{\displaystyle x=F\left(\phi,k\right)}}$ x = \incellintFk@{\phi}{k}		x = EllipticF(sin(phi), k)	x == EllipticF[\[Phi], (k)^2]	Failure	Failure	Failed [90 / 90] Result: .6791299710-.6773780507*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 1} Result: 1.016811658-.7182528229*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 2} ... skip entries to safe data	Failed [90 / 90] Result: Complex[0.6791299712710547, -0.6773780505641274] Test Values: {Rule[k, 1], Rule[x, 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[1.0168116579433883, -0.7182528227883367] Test Values: {Rule[k, 2], Rule[x, 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
22.16.E11	${\displaystyle{\displaystyle\operatorname{am}\left(x,k\right)=\phi}}$ \Jacobiamk@{x}{k} = \phi		JacobiAM(x, k) = phi	JacobiAmplitude[x, Power[k, 2]] == \[Phi]	Failure	Failure	Failed [90 / 90] Result: .2657029410-.5000000000*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 1} Result: -.6844899651-.5000000000*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 2} ... skip entries to safe data	Failed [30 / 30] Result: Complex[0.26570294146607043, -0.49999999999999994] Test Values: {Rule[k, 1], Rule[x, Rational[3, 2]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.6844899649247672, -0.49999999999999994] Test Values: {Rule[k, 2], Rule[x, Rational[3, 2]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
22.16.E12	${\displaystyle{\displaystyle\operatorname{sn}\left(x,k\right)=\sin\phi}}$ \Jacobiellsnk@{x}{k} = \sin@@{\phi}		JacobiSN(x, k) = sin(phi)	JacobiSN[x, (k)^2] == Sin[\[Phi]]	Failure	Failure	Failed [90 / 90] Result: .461679191e-1-.3375964631*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 1} Result: -.6784403409-.3375964631*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 2} ... skip entries to safe data	Failed [90 / 90] Result: Complex[0.046167919344728525, -0.33759646322287] Test Values: {Rule[k, 1], Rule[x, 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.678440340667692, -0.33759646322287] Test Values: {Rule[k, 2], Rule[x, 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
22.16.E12	${\displaystyle{\displaystyle\sin\phi=\sin\left(\operatorname{am}\left(x,k\right)\right)}}$ \sin@@{\phi} = \sin@{\Jacobiamk@{x}{k}}		sin(phi) = sin(JacobiAM(x, k))	Sin[\[Phi]] == Sin[JacobiAmplitude[x, Power[k, 2]]]	Failure	Failure	Failed [90 / 90] Result: -.461679191e-1+.3375964631*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 1} Result: .6784403409+.3375964631*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 2} ... skip entries to safe data	Failed [30 / 30] Result: Complex[-0.046167919344728525, 0.33759646322287] Test Values: {Rule[k, 1], Rule[x, Rational[3, 2]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.678440340667692, 0.33759646322287] Test Values: {Rule[k, 2], Rule[x, Rational[3, 2]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
22.16.E13	${\displaystyle{\displaystyle\operatorname{cn}\left(x,k\right)=\cos\phi}}$ \Jacobiellcnk@{x}{k} = \cos@@{\phi}		JacobiCN(x, k) = cos(phi)	JacobiCN[x, (k)^2] == Cos[\[Phi]]	Failure	Failure	Failed [90 / 90] Result: -.3054469840+.3969495503*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 1} Result: .2530246253+.3969495503*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 2} ... skip entries to safe data	Failed [90 / 90] Result: Complex[-0.3054469841149447, 0.3969495502290325] Test Values: {Rule[k, 1], Rule[x, 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[0.2530246251336542, 0.3969495502290325] Test Values: {Rule[k, 2], Rule[x, 1.5], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
22.16.E13	${\displaystyle{\displaystyle\cos\phi=\cos\left(\operatorname{am}\left(x,k\right)\right)}}$ \cos@@{\phi} = \cos@{\Jacobiamk@{x}{k}}		cos(phi) = cos(JacobiAM(x, k))	Cos[\[Phi]] == Cos[JacobiAmplitude[x, Power[k, 2]]]	Failure	Failure	Failed [90 / 90] Result: .3054469840-.3969495503*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 1} Result: -.2530246253-.3969495503*I Test Values: {phi = 1/2*3^(1/2)+1/2*I, x = 3/2, k = 2} ... skip entries to safe data	Failed [30 / 30] Result: Complex[0.3054469841149447, -0.3969495502290325] Test Values: {Rule[k, 1], Rule[x, Rational[3, 2]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Complex[-0.2530246251336542, -0.3969495502290325] Test Values: {Rule[k, 2], Rule[x, Rational[3, 2]], Rule[ϕ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} ... skip entries to safe data
22.16.E33	${\displaystyle{\displaystyle\mathrm{Z}\left(x+K|k\right)=\mathrm{Z}\left(x|k\right)-k^{2}\operatorname{sn}\left(x,k\right)\operatorname{cd}\left(x,k\right)}}$ \JacobiZetak@{x+K}{k} = \JacobiZetak@{x}{k}-k^{2}\Jacobiellsnk@{x}{k}\Jacobiellcdk@{x}{k}		JacobiZeta(x + EllipticK(k), k) = JacobiZeta(x, k)- (k)^(2)* JacobiSN(x, k)*JacobiCD(x, k)	JacobiZeta[x + EllipticK[(k)^2], k] == JacobiZeta[x, k]- (k)^(2)* JacobiSN[x, (k)^2]*JacobiCD[x, (k)^2]	Failure	Failure	Error	Failed [9 / 9] Result: Plus[-0.09234673295918805, JacobiZeta[DirectedInfinity[], 1.0]] Test Values: {Rule[k, 1], Rule[x, 1.5]} Result: Complex[-2.7319699839312124, -0.6260098794347219] Test Values: {Rule[k, 2], Rule[x, 1.5]} ... skip entries to safe data
22.16.E34	${\displaystyle{\displaystyle\mathrm{Z}\left(x+2K|k\right)=\mathrm{Z}\left(x|k\right)}}$ \JacobiZetak@{x+2K}{k} = \JacobiZetak@{x}{k}		JacobiZeta(x + 2*EllipticK(k), k) = JacobiZeta(x, k)	JacobiZeta[x + 2*EllipticK[(k)^2], k] == JacobiZeta[x, k]	Successful	Failure	-	Failed [9 / 9] Result: Plus[-0.9974949866040544, JacobiZeta[DirectedInfinity[], 1.0]] Test Values: {Rule[k, 1], Rule[x, 1.5]} Result: Complex[-0.2870190432201134, -5.437509139287473] Test Values: {Rule[k, 2], Rule[x, 1.5]} ... skip entries to safe data