EllipticLog[{Subscript[z, 1], Sqrt[Subscript[z, 1]^3 + a Subscript[z, 1]^2 + b Subscript[z, 1]]}, {a, b}] == (1/2^(1/3)) InverseWeierstrassP[(1/6) 2^(1/3) (a + 3 Subscript[z, 1]), {4^(1/3) (a^2/3 - b), (a b)/3 - (2 a^3)/27}]

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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> <mi> elog </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> , </mo> <mrow> <msqrt> <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> </mrow> </msqrt> <mo> ; </mo> <mi> a </mi> </mrow> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mroot> <mn> 2 </mn> <mn> 3 </mn> </mroot> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> &#8472; </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> <mo> &#8290; </mo> <mroot> <mn> 2 </mn> <mn> 3 </mn> </mroot> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ; </mo> <mrow> <mroot> <mn> 4 </mn> <mn> 3 </mn> </mroot> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mn> 3 </mn> </mfrac> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> , </mo> <mrow> <mfrac> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> <mn> 3 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 3 </mn> </msup> </mrow> <mn> 27 </mn> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> elog </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> CompoundExpression </ci> <apply> <power /> <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> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> a </ci> </apply> <ci> b </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> InverseWeierstrassP </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 6 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <list> <apply> <times /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> b </ci> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 27 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </list> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29