InverseJacobiSN[z, m] == (-z) EllipticLog[{1, Sqrt[a + b + 1]}, {a, b}] /; {a, b} == {(-z^2) (m + 1), m z^4} && Abs[z] < 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> sn </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> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> &#8290; </mo> <mrow> <mi> elog </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mrow> <msqrt> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ; </mo> <mi> a </mi> </mrow> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mo> { </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> } </mo> </mrow> <mo> &#10869; </mo> <mrow> <mo> { </mo> <mrow> <mrow> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mi> m </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> </mrow> <mo> } </mo> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> InverseJacobiSN </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <apply> <ci> elog </ci> <cn type='integer'> 1 </cn> <apply> <ci> CompoundExpression </ci> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> a </ci> </apply> <ci> b </ci> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <list> <ci> a </ci> <ci> b </ci> </list> <list> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <ci> m </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </list> </apply> <apply> <lt /> <apply> <abs /> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29