((m - 1) Subscript[z, 1]^2 Subscript[z, 2]^2 + 1)^2 JacobiSC[w[Subscript[z, 1]] + w[Subscript[z, 2]], m]^4 + 2 (Subscript[z, 1]^2 ((m - 1) Subscript[z, 2]^4 + ((m - 1) Subscript[z, 1]^2 + 2 (m - 2)) Subscript[z, 2]^2 - 1) - Subscript[z, 2]^2) JacobiSC[w[Subscript[z, 1]] + w[Subscript[z, 2]], m]^ 2 + (Subscript[z, 1]^2 - Subscript[z, 2]^2)^2 == 0 /; w[z] == InverseJacobiSC[z, m]

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", SubsuperscriptBox["z", "1", "2"], " ", SubsuperscriptBox["z", "2", "2"]]], "+", "1"]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["JacobiSC", "[", RowBox[List[RowBox[List[RowBox[List["w", "[", SubscriptBox["z", "1"], "]"]], "+", RowBox[List["w", "[", SubscriptBox["z", "2"], "]"]]]], ",", "m"]], "]"]], "4"]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[SubsuperscriptBox["z", "1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", SubsuperscriptBox["z", "2", "4"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m", "-", "1"]], ")"]], " ", SubsuperscriptBox["z", "1", "2"]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["m", "-", "2"]], ")"]]]]]], ")"]], " ", SubsuperscriptBox["z", "2", "2"]]], "-", "1"]], ")"]]]], "-", SubsuperscriptBox["z", "2", "2"]]], ")"]], " ", SuperscriptBox[RowBox[List["JacobiSC", "[", RowBox[List[RowBox[List[RowBox[List["w", "[", SubscriptBox["z", "1"], "]"]], "+", RowBox[List["w", "[", SubscriptBox["z", "2"], "]"]]]], ",", "m"]], "]"]], "2"]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List[SubsuperscriptBox["z", "1", "2"], "-", SubsuperscriptBox["z", "2", "2"]]], ")"]], "2"]]], "\[Equal]", "0"]], "/;", RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List["InverseJacobiSC", "[", RowBox[List["z", ",", "m"]], "]"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mrow> <mi> sc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> w </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> w </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msubsup> <mi> z </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 2 </mn> <mn> 4 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <msubsup> <mi> z </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mi> sc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> w </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> w </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msubsup> <mi> z </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> <mo> - </mo> <msubsup> <mi> z </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> w </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <msup> <mi> sc </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> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> JacobiSC </ci> <apply> <plus /> <apply> <ci> w </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> w </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <ci> m </ci> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> m </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> m </ci> <cn type='integer'> -2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <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> <apply> <power /> <apply> <ci> JacobiSC </ci> <apply> <plus /> <apply> <ci> w </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> w </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <ci> m </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </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'> 2 </cn> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <eq /> <apply> <ci> w </ci> <ci> z </ci> </apply> <apply> <ci> InverseJacobiSC </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m_", "-", "1"]], ")"]], " ", SubsuperscriptBox["z_", "1", "2"], " ", SubsuperscriptBox["z_", "2", "2"]]], "+", "1"]], ")"]], "2"], " ", SuperscriptBox[RowBox[List["JacobiSC", "[", RowBox[List[RowBox[List[RowBox[List["w", "[", SubscriptBox["z_", "1"], "]"]], "+", RowBox[List["w", "[", SubscriptBox["z_", "2"], "]"]]]], ",", "m_"]], "]"]], "4"]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[SubsuperscriptBox["z_", "1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m_", "-", "1"]], ")"]], " ", SubsuperscriptBox["z_", "2", "4"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m_", "-", "1"]], ")"]], " ", SubsuperscriptBox["z_", "1", "2"]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["m_", "-", "2"]], ")"]]]]]], ")"]], " ", SubsuperscriptBox["z_", "2", "2"]]], "-", "1"]], ")"]]]], "-", SubsuperscriptBox["z_", "2", "2"]]], ")"]], " ", SuperscriptBox[RowBox[List["JacobiSC", "[", RowBox[List[RowBox[List[RowBox[List["w", "[", SubscriptBox["z_", "1"], "]"]], "+", RowBox[List["w", "[", SubscriptBox["z_", "2"], "]"]]]], ",", "m_"]], "]"]], "2"]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List[SubsuperscriptBox["z_", "1", "2"], "-", SubsuperscriptBox["z_", "2", "2"]]], ")"]], "2"]]], "]"]], "\[RuleDelayed]", RowBox[List["0", "/;", RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List["InverseJacobiSC", "[", RowBox[List["z", ",", "m"]], "]"]]]]]]]]]]

2001-10-29