Jacobian Elliptic Functions - 22.7 Landen Transformations

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
22.7.E2 sn ⁡ ( z , k ) = ( 1 + k 1 ) ⁢ sn ⁡ ( z / ( 1 + k 1 ) , k 1 ) 1 + k 1 ⁢ sn 2 ⁡ ( z / ( 1 + k 1 ) , k 1 ) Jacobi-elliptic-sn 𝑧 𝑘 1 subscript 𝑘 1 Jacobi-elliptic-sn 𝑧 1 subscript 𝑘 1 subscript 𝑘 1 1 subscript 𝑘 1 Jacobi-elliptic-sn 2 𝑧 1 subscript 𝑘 1 subscript 𝑘 1 {\displaystyle{\displaystyle\operatorname{sn}\left(z,k\right)=\frac{(1+k_{1})% \operatorname{sn}\left(z/(1+k_{1}),k_{1}\right)}{1+k_{1}{\operatorname{sn}^{2}% }\left(z/(1+k_{1}),k_{1}\right)}}}
\Jacobiellsnk@{z}{k} = \frac{(1+k_{1})\Jacobiellsnk@{z/(1+k_{1})}{k_{1}}}{1+k_{1}\Jacobiellsnk^{2}@{z/(1+k_{1})}{k_{1}}}		 

JacobiSN(z, k) = ((1 +((1 -sqrt(1 - (k)^(2)))/(1 +sqrt(1 - (k)^(2)))))*JacobiSN(z/(1 +((1 -sqrt(1 - (k)^(2)))/(1 +sqrt(1 - (k)^(2))))), (1 -sqrt(1 - (k)^(2)))/(1 +sqrt(1 - (k)^(2)))))/(1 +((1 -sqrt(1 - (k)^(2)))/(1 +sqrt(1 - (k)^(2))))*(JacobiSN(z/(1 +((1 -sqrt(1 - (k)^(2)))/(1 +sqrt(1 - (k)^(2))))), (1 -sqrt(1 - (k)^(2)))/(1 +sqrt(1 - (k)^(2)))))^(2))

JacobiSN[z, (k)^2] == Divide[(1 +(Divide[1 -Sqrt[1 - (k)^(2)],1 +Sqrt[1 - (k)^(2)]]))*JacobiSN[z/(1 +(Divide[1 -Sqrt[1 - (k)^(2)],1 +Sqrt[1 - (k)^(2)]])), (Divide[1 -Sqrt[1 - (k)^(2)],1 +Sqrt[1 - (k)^(2)]])^2],1 +(Divide[1 -Sqrt[1 - (k)^(2)],1 +Sqrt[1 - (k)^(2)]])*(JacobiSN[z/(1 +(Divide[1 -Sqrt[1 - (k)^(2)],1 +Sqrt[1 - (k)^(2)]])), (Divide[1 -Sqrt[1 - (k)^(2)],1 +Sqrt[1 - (k)^(2)]])^2])^(2)]
Failure Aborted Successful [Tested: 21] Successful [Tested: 21]
22.7.E3 cn ⁡ ( z , k ) = cn ⁡ ( z / ( 1 + k 1 ) , k 1 ) ⁢ dn ⁡ ( z / ( 1 + k 1 ) , k 1 ) 1 + k 1 ⁢ sn 2 ⁡ ( z / ( 1 + k 1 ) , k 1 ) Jacobi-elliptic-cn 𝑧 𝑘 Jacobi-elliptic-cn 𝑧 1 subscript 𝑘 1 subscript 𝑘 1 Jacobi-elliptic-dn 𝑧 1 subscript 𝑘 1 subscript 𝑘 1 1 subscript 𝑘 1 Jacobi-elliptic-sn 2 𝑧 1 subscript 𝑘 1 subscript 𝑘 1 {\displaystyle{\displaystyle\operatorname{cn}\left(z,k\right)=\frac{% \operatorname{cn}\left(z/(1+k_{1}),k_{1}\right)\operatorname{dn}\left(z/(1+k_{% 1}),k_{1}\right)}{1+k_{1}{\operatorname{sn}^{2}}\left(z/(1+k_{1}),k_{1}\right)% }}}
\Jacobiellcnk@{z}{k} = \frac{\Jacobiellcnk@{z/(1+k_{1})}{k_{1}}\Jacobielldnk@{z/(1+k_{1})}{k_{1}}}{1+k_{1}\Jacobiellsnk^{2}@{z/(1+k_{1})}{k_{1}}}		 

JacobiCN(z, k) = (JacobiCN(z/(1 +((1 -sqrt(1 - (k)^(2)))/(1 +sqrt(1 - (k)^(2))))), (1 -sqrt(1 - (k)^(2)))/(1 +sqrt(1 - (k)^(2))))*JacobiDN(z/(1 +((1 -sqrt(1 - (k)^(2)))/(1 +sqrt(1 - (k)^(2))))), (1 -sqrt(1 - (k)^(2)))/(1 +sqrt(1 - (k)^(2)))))/(1 +((1 -sqrt(1 - (k)^(2)))/(1 +sqrt(1 - (k)^(2))))*(JacobiSN(z/(1 +((1 -sqrt(1 - (k)^(2)))/(1 +sqrt(1 - (k)^(2))))), (1 -sqrt(1 - (k)^(2)))/(1 +sqrt(1 - (k)^(2)))))^(2))

JacobiCN[z, (k)^2] == Divide[JacobiCN[z/(1 +(Divide[1 -Sqrt[1 - (k)^(2)],1 +Sqrt[1 - (k)^(2)]])), (Divide[1 -Sqrt[1 - (k)^(2)],1 +Sqrt[1 - (k)^(2)]])^2]*JacobiDN[z/(1 +(Divide[1 -Sqrt[1 - (k)^(2)],1 +Sqrt[1 - (k)^(2)]])), (Divide[1 -Sqrt[1 - (k)^(2)],1 +Sqrt[1 - (k)^(2)]])^2],1 +(Divide[1 -Sqrt[1 - (k)^(2)],1 +Sqrt[1 - (k)^(2)]])*(JacobiSN[z/(1 +(Divide[1 -Sqrt[1 - (k)^(2)],1 +Sqrt[1 - (k)^(2)]])), (Divide[1 -Sqrt[1 - (k)^(2)],1 +Sqrt[1 - (k)^(2)]])^2])^(2)]
Failure Aborted Successful [Tested: 21] Successful [Tested: 21]
22.7.E4 dn ⁡ ( z , k ) = dn 2 ⁡ ( z / ( 1 + k 1 ) , k 1 ) - ( 1 - k 1 ) 1 + k 1 - dn 2 ⁡ ( z / ( 1 + k 1 ) , k 1 ) Jacobi-elliptic-dn 𝑧 𝑘 Jacobi-elliptic-dn 2 𝑧 1 subscript 𝑘 1 subscript 𝑘 1 1 subscript 𝑘 1 1 subscript 𝑘 1 Jacobi-elliptic-dn 2 𝑧 1 subscript 𝑘 1 subscript 𝑘 1 {\displaystyle{\displaystyle\operatorname{dn}\left(z,k\right)=\frac{{% \operatorname{dn}^{2}}\left(z/(1+k_{1}),k_{1}\right)-(1-k_{1})}{1+k_{1}-{% \operatorname{dn}^{2}}\left(z/(1+k_{1}),k_{1}\right)}}}
\Jacobielldnk@{z}{k} = \frac{\Jacobielldnk^{2}@{z/(1+k_{1})}{k_{1}}-(1-k_{1})}{1+k_{1}-\Jacobielldnk^{2}@{z/(1+k_{1})}{k_{1}}}		 

JacobiDN(z, k) = ((JacobiDN(z/(1 +((1 -sqrt(1 - (k)^(2)))/(1 +sqrt(1 - (k)^(2))))), (1 -sqrt(1 - (k)^(2)))/(1 +sqrt(1 - (k)^(2)))))^(2)-(1 -((1 -sqrt(1 - (k)^(2)))/(1 +sqrt(1 - (k)^(2))))))/(1 +((1 -sqrt(1 - (k)^(2)))/(1 +sqrt(1 - (k)^(2))))- (JacobiDN(z/(1 +((1 -sqrt(1 - (k)^(2)))/(1 +sqrt(1 - (k)^(2))))), (1 -sqrt(1 - (k)^(2)))/(1 +sqrt(1 - (k)^(2)))))^(2))

JacobiDN[z, (k)^2] == Divide[(JacobiDN[z/(1 +(Divide[1 -Sqrt[1 - (k)^(2)],1 +Sqrt[1 - (k)^(2)]])), (Divide[1 -Sqrt[1 - (k)^(2)],1 +Sqrt[1 - (k)^(2)]])^2])^(2)-(1 -(Divide[1 -Sqrt[1 - (k)^(2)],1 +Sqrt[1 - (k)^(2)]])),1 +(Divide[1 -Sqrt[1 - (k)^(2)],1 +Sqrt[1 - (k)^(2)]])- (JacobiDN[z/(1 +(Divide[1 -Sqrt[1 - (k)^(2)],1 +Sqrt[1 - (k)^(2)]])), (Divide[1 -Sqrt[1 - (k)^(2)],1 +Sqrt[1 - (k)^(2)]])^2])^(2)]
Failure Aborted Successful [Tested: 21] Successful [Tested: 21]
22.7.E6 sn ⁡ ( z , k ) = ( 1 + k 2 ′ ) ⁢ sn ⁡ ( z / ( 1 + k 2 ′ ) , k 2 ) ⁢ cn ⁡ ( z / ( 1 + k 2 ′ ) , k 2 ) dn ⁡ ( z / ( 1 + k 2 ′ ) , k 2 ) Jacobi-elliptic-sn 𝑧 𝑘 1 superscript subscript 𝑘 2 ′ Jacobi-elliptic-sn 𝑧 1 superscript subscript 𝑘 2 ′ subscript 𝑘 2 Jacobi-elliptic-cn 𝑧 1 superscript subscript 𝑘 2 ′ subscript 𝑘 2 Jacobi-elliptic-dn 𝑧 1 superscript subscript 𝑘 2 ′ subscript 𝑘 2 {\displaystyle{\displaystyle\operatorname{sn}\left(z,k\right)=\frac{(1+k_{2}^{% \prime})\operatorname{sn}\left(z/(1+k_{2}^{\prime}),k_{2}\right)\operatorname{% cn}\left(z/(1+k_{2}^{\prime}),k_{2}\right)}{\operatorname{dn}\left(z/(1+k_{2}^% {\prime}),k_{2}\right)}}}
\Jacobiellsnk@{z}{k} = \frac{(1+k_{2}^{\prime})\Jacobiellsnk@{z/(1+k_{2}^{\prime})}{k_{2}}\Jacobiellcnk@{z/(1+k_{2}^{\prime})}{k_{2}}}{\Jacobielldnk@{z/(1+k_{2}^{\prime})}{k_{2}}}		 

JacobiSN(z, k) = ((1 +((1 - k)/(1 + k)))*JacobiSN(z/(1 +((1 - k)/(1 + k))), k[2])*JacobiCN(z/(1 +((1 - k)/(1 + k))), k[2]))/(JacobiDN(z/(1 +((1 - k)/(1 + k))), k[2]))

JacobiSN[z, (k)^2] == Divide[(1 +(Divide[1 - k,1 + k]))*JacobiSN[z/(1 +(Divide[1 - k,1 + k])), (Subscript[k, 2])^2]*JacobiCN[z/(1 +(Divide[1 - k,1 + k])), (Subscript[k, 2])^2],JacobiDN[z/(1 +(Divide[1 - k,1 + k])), (Subscript[k, 2])^2]]
Failure Aborted
Failed [210 / 210]
Result: .2320130981+.1889825613*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, k[2] = 1/2*3^(1/2)+1/2*I, k = 1}


Result: .4896247760+.2144288908*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, k[2] = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [210 / 210]
Result: Complex[0.23201309774017753, 0.18898256119227738]
Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[k, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.4896247756050003, 0.2144288910337357]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[k, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.7.E7 cn ⁡ ( z , k ) = ( 1 + k 2 ′ ) ⁢ ( dn 2 ⁡ ( z / ( 1 + k 2 ′ ) , k 2 ) - k 2 ′ ) k 2 2 ⁢ dn ⁡ ( z / ( 1 + k 2 ′ ) , k 2 ) Jacobi-elliptic-cn 𝑧 𝑘 1 superscript subscript 𝑘 2 ′ Jacobi-elliptic-dn 2 𝑧 1 superscript subscript 𝑘 2 ′ subscript 𝑘 2 superscript subscript 𝑘 2 ′ superscript subscript 𝑘 2 2 Jacobi-elliptic-dn 𝑧 1 superscript subscript 𝑘 2 ′ subscript 𝑘 2 {\displaystyle{\displaystyle\operatorname{cn}\left(z,k\right)=\frac{(1+k_{2}^{% \prime})({\operatorname{dn}^{2}}\left(z/(1+k_{2}^{\prime}),k_{2}\right)-k_{2}^% {\prime})}{k_{2}^{2}\operatorname{dn}\left(z/(1+k_{2}^{\prime}),k_{2}\right)}}}
\Jacobiellcnk@{z}{k} = \frac{(1+k_{2}^{\prime})(\Jacobielldnk^{2}@{z/(1+k_{2}^{\prime})}{k_{2}}-k_{2}^{\prime})}{k_{2}^{2}\Jacobielldnk@{z/(1+k_{2}^{\prime})}{k_{2}}}		 

JacobiCN(z, k) = ((1 +((1 - k)/(1 + k)))*((JacobiDN(z/(1 +((1 - k)/(1 + k))), k[2]))^(2)-((1 - k)/(1 + k))))/((k[2])^(2)*JacobiDN(z/(1 +((1 - k)/(1 + k))), k[2]))

JacobiCN[z, (k)^2] == Divide[(1 +(Divide[1 - k,1 + k]))*((JacobiDN[z/(1 +(Divide[1 - k,1 + k])), (Subscript[k, 2])^2])^(2)-(Divide[1 - k,1 + k])),(Subscript[k, 2])^(2)*JacobiDN[z/(1 +(Divide[1 - k,1 + k])), (Subscript[k, 2])^2]]
Failure Aborted
Failed [210 / 210]
Result: -.3582173507+.1286198012*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, k[2] = 1/2*3^(1/2)+1/2*I, k = 1}


Result: .5427357897+.8396234046e-1*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, k[2] = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [210 / 210]
Result: Complex[0.5228144818495482, 0.8542847397966109]
Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[k, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.6630406190754804, 0.41475216363716894]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[k, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
22.7.E8 dn ⁡ ( z , k ) = ( 1 - k 2 ′ ) ⁢ ( dn 2 ⁡ ( z / ( 1 + k 2 ′ ) , k 2 ) + k 2 ′ ) k 2 2 ⁢ dn ⁡ ( z / ( 1 + k 2 ′ ) , k 2 ) Jacobi-elliptic-dn 𝑧 𝑘 1 superscript subscript 𝑘 2 ′ Jacobi-elliptic-dn 2 𝑧 1 superscript subscript 𝑘 2 ′ subscript 𝑘 2 superscript subscript 𝑘 2 ′ superscript subscript 𝑘 2 2 Jacobi-elliptic-dn 𝑧 1 superscript subscript 𝑘 2 ′ subscript 𝑘 2 {\displaystyle{\displaystyle\operatorname{dn}\left(z,k\right)=\frac{(1-k_{2}^{% \prime})({\operatorname{dn}^{2}}\left(z/(1+k_{2}^{\prime}),k_{2}\right)+k_{2}^% {\prime})}{k_{2}^{2}\operatorname{dn}\left(z/(1+k_{2}^{\prime}),k_{2}\right)}}}
\Jacobielldnk@{z}{k} = \frac{(1-k_{2}^{\prime})(\Jacobielldnk^{2}@{z/(1+k_{2}^{\prime})}{k_{2}}+k_{2}^{\prime})}{k_{2}^{2}\Jacobielldnk@{z/(1+k_{2}^{\prime})}{k_{2}}}		 

JacobiDN(z, k) = ((1 -((1 - k)/(1 + k)))*((JacobiDN(z/(1 +((1 - k)/(1 + k))), k[2]))^(2)+((1 - k)/(1 + k))))/((k[2])^(2)*JacobiDN(z/(1 +((1 - k)/(1 + k))), k[2]))

JacobiDN[z, (k)^2] == Divide[(1 -(Divide[1 - k,1 + k]))*((JacobiDN[z/(1 +(Divide[1 - k,1 + k])), (Subscript[k, 2])^2])^(2)+(Divide[1 - k,1 + k])),(Subscript[k, 2])^(2)*JacobiDN[z/(1 +(Divide[1 - k,1 + k])), (Subscript[k, 2])^2]]
Failure Aborted
Failed [210 / 210]
Result: -.3582173507+.1286198012*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, k[2] = 1/2*3^(1/2)+1/2*I, k = 1}


Result: -.2544342076-.6669510446*I
Test Values: {z = 1/2*3^(1/2)+1/2*I, k[2] = 1/2*3^(1/2)+1/2*I, k = 2}

... skip entries to safe data
Failed [210 / 210]
Result: Complex[0.5228144818495482, 0.8542847397966109]
Test Values: {Rule[k, 1], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[k, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Complex[0.18687780488878028, -0.30624830191491115]
Test Values: {Rule[k, 2], Rule[z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[k, 2], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
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