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 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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}}}}
\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}}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | 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))
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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)]
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Failure | Aborted | Successful [Tested: 21] | Successful [Tested: 21] |
| 22.7.E3 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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}}}}
\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}}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | 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))
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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)]
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Failure | Aborted | Successful [Tested: 21] | Successful [Tested: 21] |
| 22.7.E4 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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}}}}
\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}}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | 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))
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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)]
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Failure | Aborted | Successful [Tested: 21] | Successful [Tested: 21] |
| 22.7.E6 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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}}}}
\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}}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | 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]))
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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]]
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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 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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}}}}
\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}}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | 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]))
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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]]
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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 | Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \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}}}}
\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}}} |
Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle } | 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]))
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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]]
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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 |