InverseJacobiNC[z, m] == (1/Sqrt[1 - m]) ((Sqrt[Subscript[z, 2]^2]/Subscript[z, 2]) EllipticLog[{Subscript[z, 1], Subscript[z, 2]}, {a, b}] - EllipticK[m/(m - 1)]) /; {a, b, Subscript[z, 1]} == {(2 m - 1)/(1 - m), m/(m - 1), z^2} && Subscript[z, 1]^3 + a Subscript[z, 1]^2 + b Subscript[z, 1] - Subscript[z, 2]^2 == 0 && 0 < z < 1 && m > 1

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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> <mrow> <msup> <mi> nc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <msqrt> <msubsup> <mi> z </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </msqrt> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mfrac> <mo> &#8290; </mo> <mrow> <mi> elog </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mrow> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> ; </mo> <mi> a </mi> </mrow> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mi> m </mi> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mo> { </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> <mo> , </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mrow> <mo> } </mo> </mrow> <mo> &#10869; </mo> <mrow> <mo> { </mo> <mrow> <mfrac> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> m </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> m </mi> </mrow> </mfrac> <mo> , </mo> <mfrac> <mi> m </mi> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> , </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> } </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <msubsup> <mi> z </mi> <mn> 1 </mn> <mn> 3 </mn> </msubsup> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mrow> <mo> - </mo> <msubsup> <mi> z </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mn> 0 </mn> <mo> &lt; </mo> <mi> z </mi> <mo> &lt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &gt; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> InverseJacobiNC </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> elog </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> CompoundExpression </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <ci> a </ci> </apply> <ci> b </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> EllipticK </ci> <apply> <times /> <ci> m </ci> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <list> <ci> a </ci> <ci> b </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </list> <list> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> m </ci> <apply> <power /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </list> </apply> <apply> <eq /> <apply> <plus /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <ci> a </ci> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <lt /> <cn type='integer'> 0 </cn> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <gt /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29