ArithmeticGeometricMean[-a, -b] == -ArithmeticGeometricMean[a, b] /; !Element[a, Reals] && !Element[b, Reals]

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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> <semantics> <mi> agm </mi> <annotation-xml encoding='MathML-Content'> <ci> ArithmeticGeometricMean </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mi> b </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mo> - </mo> <mrow> <semantics> <mi> agm </mi> <annotation-xml encoding='MathML-Content'> <ci> ArithmeticGeometricMean </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8713; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> b </mi> <mo> &#8713; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> ArithmeticGeometricMean </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> ArithmeticGeometricMean </ci> <ci> a </ci> <ci> b </ci> </apply> </apply> </apply> <apply> <and /> <apply> <notin /> <ci> a </ci> <reals /> </apply> <apply> <notin /> <ci> b </ci> <reals /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArithmeticGeometricMean", "[", RowBox[List[RowBox[List["-", "a_"]], ",", RowBox[List["-", "b_"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", RowBox[List["ArithmeticGeometricMean", "[", RowBox[List["a", ",", "b"]], "]"]]]], "/;", RowBox[List[RowBox[List["!", RowBox[List["a", "\[Element]", "Reals"]]]], "&&", RowBox[List["!", RowBox[List["b", "\[Element]", "Reals"]]]]]]]]]]]]

2001-10-29