Multidimensional Theta Functions - 22.2 Definitions

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
22.2#Ex1 k = θ 2 2 ⁡ ( 0 , q ) θ 3 2 ⁡ ( 0 , q ) 𝑘 Jacobi-theta 2 2 0 𝑞 Jacobi-theta 3 2 0 𝑞 {\displaystyle{\displaystyle k=\frac{{\theta_{2}^{2}}\left(0,q\right)}{{\theta% _{3}^{2}}\left(0,q\right)}}}
k = \frac{\Jacobithetaq{2}^{2}@{0}{q}}{\Jacobithetaq{3}^{2}@{0}{q}}		 

k = ((JacobiTheta2(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))^(2))/((JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))^(2))

k == Divide[(EllipticTheta[2, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]])^(2),(EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]])^(2)]
Failure Failure Error Successful [Tested: 3]
22.2#Ex2 k ′ = θ 4 2 ⁡ ( 0 , q ) θ 3 2 ⁡ ( 0 , q ) superscript 𝑘 ′ Jacobi-theta 4 2 0 𝑞 Jacobi-theta 3 2 0 𝑞 {\displaystyle{\displaystyle k^{\prime}=\frac{{\theta_{4}^{2}}\left(0,q\right)% }{{\theta_{3}^{2}}\left(0,q\right)}}}
k^{\prime} = \frac{\Jacobithetaq{4}^{2}@{0}{q}}{\Jacobithetaq{3}^{2}@{0}{q}}		 

sqrt(1 - (k)^(2)) = ((JacobiTheta4(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))^(2))/((JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))^(2))

Sqrt[1 - (k)^(2)] == Divide[(EllipticTheta[4, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]])^(2),(EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]])^(2)]
Failure Failure Error Successful [Tested: 3]
22.2#Ex3 K ⁡ ( k ) = π 2 ⁢ θ 3 2 ⁡ ( 0 , q ) complete-elliptic-integral-first-kind-K 𝑘 𝜋 2 Jacobi-theta 3 2 0 𝑞 {\displaystyle{\displaystyle K\left(k\right)=\frac{\pi}{2}{\theta_{3}^{2}}% \left(0,q\right)}}
\compellintKk@{k} = \frac{\pi}{2}\Jacobithetaq{3}^{2}@{0}{q}		 

EllipticK(k) = (Pi)/(2)*(JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))^(2)

EllipticK[(k)^2] == Divide[Pi,2]*(EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]])^(2)
Failure Failure Error
Failed [1 / 3]
Result: DirectedInfinity[]
Test Values: {Rule[k, 1]}

22.2.E3 ζ = π ⁢ z 2 ⁢ K ⁡ ( k ) 𝜁 𝜋 𝑧 2 complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle\zeta=\frac{\pi z}{2K\left(k\right)}}}
\zeta = \frac{\pi z}{2\compellintKk@{k}}		 

zeta = (Pi*z)/(2*EllipticK(k))

\[Zeta] == Divide[Pi*z,2*EllipticK[(k)^2]]
Failure Failure Error
Failed [210 / 210]
Result: Complex[0.8660254037844387, 0.49999999999999994]
Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.7059984047169785, -0.6365247818792681]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.2.E4 sn ⁡ ( z , k ) = θ 3 ⁡ ( 0 , q ) θ 2 ⁡ ( 0 , q ) ⁢ θ 1 ⁡ ( ζ , q ) θ 4 ⁡ ( ζ , q ) Jacobi-elliptic-sn 𝑧 𝑘 Jacobi-theta 3 0 𝑞 Jacobi-theta 2 0 𝑞 Jacobi-theta 1 𝜁 𝑞 Jacobi-theta 4 𝜁 𝑞 {\displaystyle{\displaystyle\operatorname{sn}\left(z,k\right)=\frac{\theta_{3}% \left(0,q\right)}{\theta_{2}\left(0,q\right)}\frac{\theta_{1}\left(\zeta,q% \right)}{\theta_{4}\left(\zeta,q\right)}}}
\Jacobiellsnk@{z}{k} = \frac{\Jacobithetaq{3}@{0}{q}}{\Jacobithetaq{2}@{0}{q}}\frac{\Jacobithetaq{1}@{\zeta}{q}}{\Jacobithetaq{4}@{\zeta}{q}}		 

JacobiSN(z, k) = (JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta2(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta1(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta4(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))

JacobiSN[z, (k)^2] == Divide[EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[2, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[1, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[4, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]
Failure Failure Error
Failed [140 / 210]
Result: Complex[0.1017958925630662, 9.78035129055685*^-4]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.08293092681074243, -0.5359189266558633]
Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.2.E4 θ 3 ⁡ ( 0 , q ) θ 2 ⁡ ( 0 , q ) ⁢ θ 1 ⁡ ( ζ , q ) θ 4 ⁡ ( ζ , q ) = 1 ns ⁡ ( z , k ) Jacobi-theta 3 0 𝑞 Jacobi-theta 2 0 𝑞 Jacobi-theta 1 𝜁 𝑞 Jacobi-theta 4 𝜁 𝑞 1 Jacobi-elliptic-ns 𝑧 𝑘 {\displaystyle{\displaystyle\frac{\theta_{3}\left(0,q\right)}{\theta_{2}\left(% 0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q\right% )}=\frac{1}{\operatorname{ns}\left(z,k\right)}}}
\frac{\Jacobithetaq{3}@{0}{q}}{\Jacobithetaq{2}@{0}{q}}\frac{\Jacobithetaq{1}@{\zeta}{q}}{\Jacobithetaq{4}@{\zeta}{q}} = \frac{1}{\Jacobiellnsk@{z}{k}}		 

(JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta2(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta1(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta4(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k)))) = (1)/(JacobiNS(z, k))

Divide[EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[2, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[1, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[4, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]] == Divide[1,JacobiNS[z, (k)^2]]
Failure Failure Error
Failed [140 / 210]
Result: Complex[-0.1017958925630662, -9.780351290556814*^-4]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.08293092681074243, 0.5359189266558634]
Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.2.E5 cn ⁡ ( z , k ) = θ 4 ⁡ ( 0 , q ) θ 2 ⁡ ( 0 , q ) ⁢ θ 2 ⁡ ( ζ , q ) θ 4 ⁡ ( ζ , q ) Jacobi-elliptic-cn 𝑧 𝑘 Jacobi-theta 4 0 𝑞 Jacobi-theta 2 0 𝑞 Jacobi-theta 2 𝜁 𝑞 Jacobi-theta 4 𝜁 𝑞 {\displaystyle{\displaystyle\operatorname{cn}\left(z,k\right)=\frac{\theta_{4}% \left(0,q\right)}{\theta_{2}\left(0,q\right)}\frac{\theta_{2}\left(\zeta,q% \right)}{\theta_{4}\left(\zeta,q\right)}}}
\Jacobiellcnk@{z}{k} = \frac{\Jacobithetaq{4}@{0}{q}}{\Jacobithetaq{2}@{0}{q}}\frac{\Jacobithetaq{2}@{\zeta}{q}}{\Jacobithetaq{4}@{\zeta}{q}}		 

JacobiCN(z, k) = (JacobiTheta4(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta2(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta2(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta4(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))

JacobiCN[z, (k)^2] == Divide[EllipticTheta[4, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[2, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[2, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[4, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]
Failure Failure Error
Failed [140 / 210]
Result: Complex[-0.08257811120249814, 0.0027270134984790223]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.13231049687767538, 0.2777560839806882]
Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.2.E5 θ 4 ⁡ ( 0 , q ) θ 2 ⁡ ( 0 , q ) ⁢ θ 2 ⁡ ( ζ , q ) θ 4 ⁡ ( ζ , q ) = 1 nc ⁡ ( z , k ) Jacobi-theta 4 0 𝑞 Jacobi-theta 2 0 𝑞 Jacobi-theta 2 𝜁 𝑞 Jacobi-theta 4 𝜁 𝑞 1 Jacobi-elliptic-nc 𝑧 𝑘 {\displaystyle{\displaystyle\frac{\theta_{4}\left(0,q\right)}{\theta_{2}\left(% 0,q\right)}\frac{\theta_{2}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q\right% )}=\frac{1}{\operatorname{nc}\left(z,k\right)}}}
\frac{\Jacobithetaq{4}@{0}{q}}{\Jacobithetaq{2}@{0}{q}}\frac{\Jacobithetaq{2}@{\zeta}{q}}{\Jacobithetaq{4}@{\zeta}{q}} = \frac{1}{\Jacobiellnck@{z}{k}}		 

(JacobiTheta4(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta2(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta2(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta4(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k)))) = (1)/(JacobiNC(z, k))

Divide[EllipticTheta[4, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[2, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[2, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[4, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]] == Divide[1,JacobiNC[z, (k)^2]]
Failure Failure Error
Failed [140 / 210]
Result: Complex[0.08257811120249814, -0.002727013498479031]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.13231049687767538, -0.2777560839806882]
Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.2.E6 dn ⁡ ( z , k ) = θ 4 ⁡ ( 0 , q ) θ 3 ⁡ ( 0 , q ) ⁢ θ 3 ⁡ ( ζ , q ) θ 4 ⁡ ( ζ , q ) Jacobi-elliptic-dn 𝑧 𝑘 Jacobi-theta 4 0 𝑞 Jacobi-theta 3 0 𝑞 Jacobi-theta 3 𝜁 𝑞 Jacobi-theta 4 𝜁 𝑞 {\displaystyle{\displaystyle\operatorname{dn}\left(z,k\right)=\frac{\theta_{4}% \left(0,q\right)}{\theta_{3}\left(0,q\right)}\frac{\theta_{3}\left(\zeta,q% \right)}{\theta_{4}\left(\zeta,q\right)}}}
\Jacobielldnk@{z}{k} = \frac{\Jacobithetaq{4}@{0}{q}}{\Jacobithetaq{3}@{0}{q}}\frac{\Jacobithetaq{3}@{\zeta}{q}}{\Jacobithetaq{4}@{\zeta}{q}}		 

JacobiDN(z, k) = (JacobiTheta4(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta3(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta4(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))

JacobiDN[z, (k)^2] == Divide[EllipticTheta[4, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[3, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[4, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]
Failure Failure Error
Failed [140 / 210]
Result: Complex[-0.11217526698173597, -1.5044574583405517]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-1.6119897435833945, -2.3508894631681736]
Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.2.E6 θ 4 ⁡ ( 0 , q ) θ 3 ⁡ ( 0 , q ) ⁢ θ 3 ⁡ ( ζ , q ) θ 4 ⁡ ( ζ , q ) = 1 nd ⁡ ( z , k ) Jacobi-theta 4 0 𝑞 Jacobi-theta 3 0 𝑞 Jacobi-theta 3 𝜁 𝑞 Jacobi-theta 4 𝜁 𝑞 1 Jacobi-elliptic-nd 𝑧 𝑘 {\displaystyle{\displaystyle\frac{\theta_{4}\left(0,q\right)}{\theta_{3}\left(% 0,q\right)}\frac{\theta_{3}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q\right% )}=\frac{1}{\operatorname{nd}\left(z,k\right)}}}
\frac{\Jacobithetaq{4}@{0}{q}}{\Jacobithetaq{3}@{0}{q}}\frac{\Jacobithetaq{3}@{\zeta}{q}}{\Jacobithetaq{4}@{\zeta}{q}} = \frac{1}{\Jacobiellndk@{z}{k}}		 

(JacobiTheta4(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta3(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta4(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k)))) = (1)/(JacobiND(z, k))

Divide[EllipticTheta[4, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[3, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[4, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]] == Divide[1,JacobiND[z, (k)^2]]
Failure Failure Error
Failed [140 / 210]
Result: Complex[0.112175266981736, 1.5044574583405517]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.6119897435833943, 2.350889463168173]
Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.2.E7 sd ⁡ ( z , k ) = θ 3 2 ⁡ ( 0 , q ) θ 2 ⁡ ( 0 , q ) ⁢ θ 4 ⁡ ( 0 , q ) ⁢ θ 1 ⁡ ( ζ , q ) θ 3 ⁡ ( ζ , q ) Jacobi-elliptic-sd 𝑧 𝑘 Jacobi-theta 3 2 0 𝑞 Jacobi-theta 2 0 𝑞 Jacobi-theta 4 0 𝑞 Jacobi-theta 1 𝜁 𝑞 Jacobi-theta 3 𝜁 𝑞 {\displaystyle{\displaystyle\operatorname{sd}\left(z,k\right)=\frac{{\theta_{3% }^{2}}\left(0,q\right)}{\theta_{2}\left(0,q\right)\theta_{4}\left(0,q\right)}% \frac{\theta_{1}\left(\zeta,q\right)}{\theta_{3}\left(\zeta,q\right)}}}
\Jacobiellsdk@{z}{k} = \frac{\Jacobithetaq{3}^{2}@{0}{q}}{\Jacobithetaq{2}@{0}{q}\Jacobithetaq{4}@{0}{q}}\frac{\Jacobithetaq{1}@{\zeta}{q}}{\Jacobithetaq{3}@{\zeta}{q}}		 

JacobiSD(z, k) = ((JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))^(2))/(JacobiTheta2(0, exp(- Pi*EllipticCK(k)/EllipticK(k)))*JacobiTheta4(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta1(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta3(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))

JacobiSD[z, (k)^2] == Divide[(EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]])^(2),EllipticTheta[2, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]*EllipticTheta[4, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[1, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[3, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]
Failure Failure Error
Failed [138 / 210]
Result: Complex[-0.10264566281694597, 1.7190366283522571]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.005017214212665183, 0.8218706074973681]
Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.2.E7 θ 3 2 ⁡ ( 0 , q ) θ 2 ⁡ ( 0 , q ) ⁢ θ 4 ⁡ ( 0 , q ) ⁢ θ 1 ⁡ ( ζ , q ) θ 3 ⁡ ( ζ , q ) = 1 ds ⁡ ( z , k ) Jacobi-theta 3 2 0 𝑞 Jacobi-theta 2 0 𝑞 Jacobi-theta 4 0 𝑞 Jacobi-theta 1 𝜁 𝑞 Jacobi-theta 3 𝜁 𝑞 1 Jacobi-elliptic-ds 𝑧 𝑘 {\displaystyle{\displaystyle\frac{{\theta_{3}^{2}}\left(0,q\right)}{\theta_{2}% \left(0,q\right)\theta_{4}\left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right% )}{\theta_{3}\left(\zeta,q\right)}=\frac{1}{\operatorname{ds}\left(z,k\right)}}}
\frac{\Jacobithetaq{3}^{2}@{0}{q}}{\Jacobithetaq{2}@{0}{q}\Jacobithetaq{4}@{0}{q}}\frac{\Jacobithetaq{1}@{\zeta}{q}}{\Jacobithetaq{3}@{\zeta}{q}} = \frac{1}{\Jacobielldsk@{z}{k}}		 

((JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))^(2))/(JacobiTheta2(0, exp(- Pi*EllipticCK(k)/EllipticK(k)))*JacobiTheta4(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta1(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta3(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k)))) = (1)/(JacobiDS(z, k))

Divide[(EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]])^(2),EllipticTheta[2, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]*EllipticTheta[4, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[1, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[3, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]] == Divide[1,JacobiDS[z, (k)^2]]
Failure Failure Error
Failed [138 / 210]
Result: Complex[0.1026456628169461, -1.7190366283522573]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.005017214212665148, -0.8218706074973681]
Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.2.E8 cd ⁡ ( z , k ) = θ 3 ⁡ ( 0 , q ) θ 2 ⁡ ( 0 , q ) ⁢ θ 2 ⁡ ( ζ , q ) θ 3 ⁡ ( ζ , q ) Jacobi-elliptic-cd 𝑧 𝑘 Jacobi-theta 3 0 𝑞 Jacobi-theta 2 0 𝑞 Jacobi-theta 2 𝜁 𝑞 Jacobi-theta 3 𝜁 𝑞 {\displaystyle{\displaystyle\operatorname{cd}\left(z,k\right)=\frac{\theta_{3}% \left(0,q\right)}{\theta_{2}\left(0,q\right)}\frac{\theta_{2}\left(\zeta,q% \right)}{\theta_{3}\left(\zeta,q\right)}}}
\Jacobiellcdk@{z}{k} = \frac{\Jacobithetaq{3}@{0}{q}}{\Jacobithetaq{2}@{0}{q}}\frac{\Jacobithetaq{2}@{\zeta}{q}}{\Jacobithetaq{3}@{\zeta}{q}}		 

JacobiCD(z, k) = (JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta2(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta2(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta3(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))

JacobiCD[z, (k)^2] == Divide[EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[2, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[2, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[3, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]
Failure Failure Error
Failed [140 / 210]
Result: Complex[-0.23207264303523145, 2.174081147069575]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.3131092646447684, 1.178043032175558]
Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.2.E8 θ 3 ⁡ ( 0 , q ) θ 2 ⁡ ( 0 , q ) ⁢ θ 2 ⁡ ( ζ , q ) θ 3 ⁡ ( ζ , q ) = 1 dc ⁡ ( z , k ) Jacobi-theta 3 0 𝑞 Jacobi-theta 2 0 𝑞 Jacobi-theta 2 𝜁 𝑞 Jacobi-theta 3 𝜁 𝑞 1 Jacobi-elliptic-dc 𝑧 𝑘 {\displaystyle{\displaystyle\frac{\theta_{3}\left(0,q\right)}{\theta_{2}\left(% 0,q\right)}\frac{\theta_{2}\left(\zeta,q\right)}{\theta_{3}\left(\zeta,q\right% )}=\frac{1}{\operatorname{dc}\left(z,k\right)}}}
\frac{\Jacobithetaq{3}@{0}{q}}{\Jacobithetaq{2}@{0}{q}}\frac{\Jacobithetaq{2}@{\zeta}{q}}{\Jacobithetaq{3}@{\zeta}{q}} = \frac{1}{\Jacobielldck@{z}{k}}		 

(JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta2(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta2(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta3(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k)))) = (1)/(JacobiDC(z, k))

Divide[EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[2, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[2, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[3, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]] == Divide[1,JacobiDC[z, (k)^2]]
Failure Failure Error
Failed [140 / 210]
Result: Complex[0.23207264303523142, -2.174081147069575]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.3131092646447683, -1.178043032175558]
Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.2.E9 sc ⁡ ( z , k ) = θ 3 ⁡ ( 0 , q ) θ 4 ⁡ ( 0 , q ) ⁢ θ 1 ⁡ ( ζ , q ) θ 2 ⁡ ( ζ , q ) Jacobi-elliptic-sc 𝑧 𝑘 Jacobi-theta 3 0 𝑞 Jacobi-theta 4 0 𝑞 Jacobi-theta 1 𝜁 𝑞 Jacobi-theta 2 𝜁 𝑞 {\displaystyle{\displaystyle\operatorname{sc}\left(z,k\right)=\frac{\theta_{3}% \left(0,q\right)}{\theta_{4}\left(0,q\right)}\frac{\theta_{1}\left(\zeta,q% \right)}{\theta_{2}\left(\zeta,q\right)}}}
\Jacobiellsck@{z}{k} = \frac{\Jacobithetaq{3}@{0}{q}}{\Jacobithetaq{4}@{0}{q}}\frac{\Jacobithetaq{1}@{\zeta}{q}}{\Jacobithetaq{2}@{\zeta}{q}}		 

JacobiSC(z, k) = (JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta4(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta1(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta2(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))

JacobiSC[z, (k)^2] == Divide[EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[4, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[1, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[2, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]
Failure Failure Error
Failed [140 / 210]
Result: Complex[0.2180891710993932, -0.009050644683206828]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[-0.13880139985550538, -0.6261898650931494]
Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.2.E9 θ 3 ⁡ ( 0 , q ) θ 4 ⁡ ( 0 , q ) ⁢ θ 1 ⁡ ( ζ , q ) θ 2 ⁡ ( ζ , q ) = 1 cs ⁡ ( z , k ) Jacobi-theta 3 0 𝑞 Jacobi-theta 4 0 𝑞 Jacobi-theta 1 𝜁 𝑞 Jacobi-theta 2 𝜁 𝑞 1 Jacobi-elliptic-cs 𝑧 𝑘 {\displaystyle{\displaystyle\frac{\theta_{3}\left(0,q\right)}{\theta_{4}\left(% 0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{2}\left(\zeta,q\right% )}=\frac{1}{\operatorname{cs}\left(z,k\right)}}}
\frac{\Jacobithetaq{3}@{0}{q}}{\Jacobithetaq{4}@{0}{q}}\frac{\Jacobithetaq{1}@{\zeta}{q}}{\Jacobithetaq{2}@{\zeta}{q}} = \frac{1}{\Jacobiellcsk@{z}{k}}		 

(JacobiTheta3(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta4(0, exp(- Pi*EllipticCK(k)/EllipticK(k))))*(JacobiTheta1(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k))))/(JacobiTheta2(zeta, exp(- Pi*EllipticCK(k)/EllipticK(k)))) = (1)/(JacobiCS(z, k))

Divide[EllipticTheta[3, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[4, 0, Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]]*Divide[EllipticTheta[1, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]],EllipticTheta[2, \[Zeta], Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]]]] == Divide[1,JacobiCS[z, (k)^2]]
Failure Failure Error
Failed [140 / 210]
Result: Complex[-0.2180891710993933, 0.009050644683206842]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.13880139985550533, 0.6261898650931494]
Test Values: {Rule[k, 3], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ζ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.2.E11 p ⁣ q ⁡ ( z , k ) = θ p ⁡ ( z | τ ) / θ q ⁡ ( z | τ ) abstract-Jacobi-elliptic p q 𝑧 𝑘 Jacobi-theta-tau 𝑝 𝑧 𝜏 Jacobi-theta-tau 𝑞 𝑧 𝜏 {\displaystyle{\displaystyle\operatorname{pq}\left(z,k\right)=\ifrac{\theta_{p% }\left(z\middle|\tau\right)}{\theta_{q}\left(z\middle|\tau\right)}}}
\genJacobiellk{p}{q}@{z}{k} = \ifrac{\Jacobithetatau{p}@{z}{\tau}}{\Jacobithetatau{q}@{z}{\tau}}		 

genJacobiellk(p)*(exp(- Pi*EllipticCK(k)/EllipticK(k)))* z*k = (JacobiThetap(z,exp(I*Pi*tau)))/(JacobiThetaexp(- Pi*EllipticCK(k)/EllipticK(k))(z,exp(I*Pi*tau)))

genJacobiellk[p]*(Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]])* z*k == Divide[EllipticTheta[p, z, Exp[I*Pi*(\[Tau])]],EllipticTheta[Exp[- Pi*EllipticK[1-(k)^2]/EllipticK[(k)^2]], z, Exp[I*Pi*(\[Tau])]]]
Failure Failure Error
Failed [300 / 300]
Result: Plus[Times[Complex[0.5000000000000001, 0.8660254037844386], genJacobiellk], Times[Complex[-0.31964140165035193, 0.682988488811487], EllipticTheta[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Complex[0.8660254037844387, 0.49999999999999994], Complex[-0.1897367196265697, 0.08493465422971205]]]]
Test Values: {Rule[k, 1], Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[Times[Complex[0.26976733074627424, -0.3419272748333145], genJacobiellk], Times[-1.0, EllipticTheta[Power[E, Times[Complex[0, Rational[1, 6]], Pi]], Complex[0.8660254037844387, 0.49999999999999994], Complex[-0.1897367196265697, 0.08493465422971205]], Power[EllipticTheta[Power[E, Times[-1, Pi, EllipticK[-3], Power[EllipticK[4], -1]]], Complex[0.8660254037844387, 0.49999999999999994], Complex[-0.1897367196265697, 0.08493465422971205]], -1]]]
Test Values: {Rule[k, 2], Rule[p, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.2.E12 τ = i ⁢ K ′ ⁡ ( k ) / K ⁡ ( k ) 𝜏 imaginary-unit complementary-complete-elliptic-integral-first-kind-K 𝑘 complete-elliptic-integral-first-kind-K 𝑘 {\displaystyle{\displaystyle\tau=\ifrac{\mathrm{i}{K^{\prime}}\left(k\right)}{% K\left(k\right)}}}
\tau = \ifrac{\iunit\ccompellintKk@{k}}{\compellintKk@{k}}		 

tau = (I*EllipticCK(k))/(EllipticK(k))

\[Tau] == Divide[I*EllipticK[1-(k)^2],EllipticK[(k)^2]]
Failure Failure Error
Failed [30 / 30]
Result: Complex[0.8660254037844387, 0.49999999999999994]
Test Values: {Rule[k, 1], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[1.4867361401447923, 0.0147898206680519]
Test Values: {Rule[k, 2], Rule[τ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
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