EllipticExp[z, {a, b}] == {x, y} /; (z == EllipticLog[{x, y}, {a, b}] /; y^2 - x (x^2 + a x + b) == 0)

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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> <mi> eexp </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> ; </mo> <mi> a </mi> </mrow> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mo> { </mo> <mrow> <mi> x </mi> <mo> , </mo> <mi> y </mi> </mrow> <mo> } </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> &#10869; </mo> <mrow> <mi> elog </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> , </mo> <mrow> <mi> y </mi> <mo> ; </mo> <mi> a </mi> </mrow> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mi> x </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> + </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mn> 0 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> eexp </ci> <apply> <ci> CompoundExpression </ci> <ci> z </ci> <ci> a </ci> </apply> <ci> b </ci> </apply> <list> <ci> x </ci> <ci> y </ci> </list> </apply> <apply> <ci> Condition </ci> <apply> <eq /> <ci> z </ci> <apply> <ci> elog </ci> <ci> x </ci> <apply> <ci> CompoundExpression </ci> <ci> y </ci> <ci> a </ci> </apply> <ci> b </ci> </apply> </apply> <apply> <eq /> <apply> <plus /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> x </ci> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <ci> a </ci> <ci> x </ci> </apply> <ci> b </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29