EllipticPi[Conjugate[n], Conjugate[z], Conjugate[m]] == Conjugate[EllipticPi[n, z, m]] /; !(Element[m, Reals] && m > 1 && Element[n, Reals] && n > 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> <mi> &#928; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mover> <mi> n </mi> <mo> _ </mo> </mover> <mo> ; </mo> <mrow> <mover> <mi> z </mi> <mo> _ </mo> </mover> <mo> &#10072; </mo> <mover> <mi> m </mi> <mo> _ </mo> </mover> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mover> <semantics> <mrow> <mi> &#928; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> ; </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[CapitalPi]&quot;, &quot;(&quot;, RowBox[List[&quot;n&quot;, &quot;;&quot;, RowBox[List[&quot;z&quot;, &quot;\[VerticalSeparator]&quot;, &quot;m&quot;]]]], &quot;)&quot;]], BernoulliB] </annotation> </semantics> <mo> _ </mo> </mover> </mrow> <mo> /; </mo> <mrow> <mo> &#172; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> m </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[List[], Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &gt; </mo> <mn> 1 </mn> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[List[], Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &gt; </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> EllipticPi </ci> <apply> <ci> OverBar </ci> <ci> n </ci> </apply> <apply> <ci> OverBar </ci> <ci> z </ci> </apply> <apply> <ci> OverBar </ci> <ci> m </ci> </apply> </apply> <apply> <ci> OverBar </ci> <apply> <ci> BernoulliB </ci> <apply> <ci> EllipticPi </ci> <ci> n </ci> <ci> z </ci> <ci> m </ci> </apply> </apply> </apply> </apply> <apply> <not /> <apply> <and /> <apply> <in /> <ci> m </ci> <reals /> </apply> <apply> <gt /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> <apply> <in /> <ci> n </ci> <reals /> </apply> <apply> <gt /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29