Limit[ArithmeticGeometricMean[a + I \[Epsilon], 1], \[Epsilon] -> Plus[0]] == ArithmeticGeometricMean[a, 1] /; a < 0

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Limit", "[", RowBox[List[RowBox[List["ArithmeticGeometricMean", "[", RowBox[List[RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "\[Epsilon]"]]]], ",", "1"]], "]"]], ",", RowBox[List["\[Epsilon]", "\[Rule]", RowBox[List["+", "0"]]]]]], "]"]], "\[Equal]", RowBox[List["ArithmeticGeometricMean", "[", RowBox[List["a", ",", "1"]], "]"]]]], "/;", " ", RowBox[List["a", "<", "0"]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <munder> <mi> lim </mi> <mrow> <mi> &#1013; </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mrow> <mo> + </mo> <mn> 0 </mn> </mrow> </mrow> </munder> <mo> &#8290; </mo> <mtext> &#8201; </mtext> <mrow> <semantics> <mi> agm </mi> <annotation-xml encoding='MathML-Content'> <ci> ArithmeticGeometricMean </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#1013; </mi> </mrow> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </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> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> a </mi> <mo> &lt; </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <limit /> <bvar> <ci> &#1013; </ci> </bvar> <condition> <apply> <tendsto /> <ci> &#1013; </ci> <apply> <plus /> <cn type='integer'> 0 </cn> </apply> </apply> </condition> <apply> <ci> ArithmeticGeometricMean </ci> <apply> <plus /> <ci> a </ci> <apply> <times /> <imaginaryi /> <ci> &#1013; </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> ArithmeticGeometricMean </ci> <ci> a </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <lt /> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Limit", "[", RowBox[List[RowBox[List["ArithmeticGeometricMean", "[", RowBox[List[RowBox[List["a_", "+", RowBox[List["\[ImaginaryI]", " ", "\[Epsilon]_"]]]], ",", "1"]], "]"]], ",", RowBox[List["\[Epsilon]_", "\[Rule]", RowBox[List["+", "0"]]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["ArithmeticGeometricMean", "[", RowBox[List["a", ",", "1"]], "]"]], "/;", RowBox[List["a", "<", "0"]]]]]]]]

2001-10-29