EllipticK[1 - z^2]/EllipticK[z^2] == Sqrt[11] /; z == (1/2) Sqrt[(32 3^(2/3) + 3 (9 + 7 Sqrt[33])^(1/3) - 4 3^(1/3) (9 + 7 Sqrt[33])^(2/3))/(6 (9 + 7 Sqrt[33])^(1/3) + Sqrt[3 (-32 3^(2/3) (9 + 7 Sqrt[33])^(1/3) + 9 (9 + 7 Sqrt[33])^(2/3) + 4 3^(1/3) (9 + 7 Sqrt[33]))])]

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mfrac> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> </mfrac> <mo> &#10869; </mo> <msqrt> <mn> 11 </mn> </msqrt> </mrow> <mo> /; </mo> <mrow> <mi> z </mi> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8730; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 32 </mn> <mo> &#8290; </mo> <msup> <mn> 3 </mn> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 9 </mn> <mo> + </mo> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <msqrt> <mn> 33 </mn> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mn> 3 </mn> <mrow> <mn> 1 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 9 </mn> <mo> + </mo> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <msqrt> <mn> 33 </mn> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 9 </mn> <mo> + </mo> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <msqrt> <mn> 33 </mn> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> </mrow> <mo> + </mo> <msqrt> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 32 </mn> </mrow> <mo> &#8290; </mo> <msup> <mn> 3 </mn> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 9 </mn> <mo> + </mo> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <msqrt> <mn> 33 </mn> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mn> 9 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 9 </mn> <mo> + </mo> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <msqrt> <mn> 33 </mn> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mn> 3 </mn> <mrow> <mn> 1 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 9 </mn> <mo> + </mo> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <msqrt> <mn> 33 </mn> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <ci> EllipticK </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <ci> EllipticK </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> 11 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <eq /> <ci> z </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <root /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 32 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 2 <sep /> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 9 </cn> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <cn type='integer'> 33 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 9 </cn> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <cn type='integer'> 33 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 2 <sep /> 3 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 9 </cn> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <cn type='integer'> 33 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -32 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 9 </cn> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <cn type='integer'> 33 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 9 </cn> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <cn type='integer'> 33 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 2 <sep /> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <plus /> <cn type='integer'> 9 </cn> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <cn type='integer'> 33 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2002-12-18