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ArithmeticGeometricMean






Mathematica Notation

Traditional Notation









Elliptic Functions > ArithmeticGeometricMean[a,b] > Limit representations





http://functions.wolfram.com/09.54.09.0001.01









  


  










Input Form





ArithmeticGeometricMean[a, b] == Limit[Subscript[a, n], n -> Infinity] == Limit[Subscript[b, n], n -> Infinity] /; Inequality[Subscript[a, 0], Equal, a, Greater, Subscript[b, 0], Equal, b, Greater, 0] && Subscript[a, n + 1] == (1/2) (Subscript[a, n] + Subscript[b, n]) == ArithmeticGeometricMean[Subscript[a, 0], Subscript[b, 0]] EllipticTheta[3, 0, z^2^(n + 1)]^2 && Subscript[b, n + 1] == Sqrt[Subscript[a, n] Subscript[b, n]] == ArithmeticGeometricMean[Subscript[a, 0], Subscript[b, 0]] EllipticTheta[4, 0, z^2^(n + 1)]^2 && z == EllipticNomeQ[1 - (Subscript[b, 0]/Subscript[a, 0])^2]










Standard Form





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MathML Form







<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> <mi> a </mi> <mo> , </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> n </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> </munder> <mo> &#8290; </mo> <mtext> &#8201; </mtext> <msub> <mi> a </mi> <mi> n </mi> </msub> </mrow> <mo> &#10869; </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> n </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> </munder> <mo> &#8290; </mo> <mtext> &#8201; </mtext> <msub> <mi> b </mi> <mi> n </mi> </msub> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> &#10869; </mo> <mi> a </mi> <mo> &gt; </mo> <msub> <mi> b </mi> <mn> 0 </mn> </msub> <mo> &#10869; </mo> <mi> b </mi> <mo> &gt; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> a </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mi> n </mi> </msub> <mo> + </mo> <msub> <mi> b </mi> <mi> n </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <semantics> <mi> agm </mi> <annotation-xml encoding='MathML-Content'> <ci> ArithmeticGeometricMean </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <msub> <mi> b </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <msub> <mi> &#977; </mi> <mn> 3 </mn> </msub> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <msup> <mi> z </mi> <msup> <mn> 2 </mn> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> b </mi> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> &#10869; </mo> <msqrt> <mrow> <msub> <mi> a </mi> <mi> n </mi> </msub> <mo> &#8290; </mo> <msub> <mi> b </mi> <mi> n </mi> </msub> </mrow> </msqrt> <mo> &#10869; </mo> <mrow> <mrow> <semantics> <mi> agm </mi> <annotation-xml encoding='MathML-Content'> <ci> ArithmeticGeometricMean </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> , </mo> <msub> <mi> b </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <msub> <mi> &#977; </mi> <mn> 4 </mn> </msub> <mo> ( </mo> <mrow> <mn> 0 </mn> <mo> , </mo> <msup> <mi> z </mi> <msup> <mn> 2 </mn> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> z </mi> <mo> &#10869; </mo> <mrow> <semantics> <mi> q </mi> <annotation-xml encoding='MathML-Content'> <ci> EllipticNomeQ </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <msub> <mi> b </mi> <mn> 0 </mn> </msub> <msub> <mi> a </mi> <mn> 0 </mn> </msub> </mfrac> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> ArithmeticGeometricMean </ci> <ci> a </ci> <ci> b </ci> </apply> <apply> <limit /> <bvar> <ci> n </ci> </bvar> <condition> <apply> <tendsto /> <ci> n </ci> <infinity /> </apply> </condition> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> n </ci> </apply> </apply> <apply> <limit /> <bvar> <ci> n </ci> </bvar> <condition> <apply> <tendsto /> <ci> n </ci> <infinity /> </apply> </condition> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> n </ci> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Inequality </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <eq /> <ci> a </ci> <gt /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 0 </cn> </apply> <eq /> <ci> b </ci> <gt /> <cn type='integer'> 0 </cn> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> a </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> n </ci> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> n </ci> </apply> </apply> </apply> <apply> <times /> <apply> <ci> ArithmeticGeometricMean </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <power /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 3 </cn> <cn type='integer'> 0 </cn> <apply> <power /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> b </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> n </ci> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> n </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <apply> <ci> ArithmeticGeometricMean </ci> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <power /> <apply> <ci> EllipticTheta </ci> <cn type='integer'> 4 </cn> <cn type='integer'> 0 </cn> <apply> <power /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <eq /> <ci> z </ci> <apply> <ci> EllipticNomeQ </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 0 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArithmeticGeometricMean", "[", RowBox[List["a", ",", "b"]], "]"]], "\[Equal]", RowBox[List["Limit", "[", RowBox[List[SubscriptBox["a", "n"], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]], "\[Equal]", RowBox[List["Limit", "[", RowBox[List[SubscriptBox["b", "n"], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "0"], "\[Equal]", "a", ">", SubscriptBox["b", "0"], "\[Equal]", "b", ">", "0"]], "&&", RowBox[List[SubscriptBox["a", RowBox[List["n", "+", "1"]]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[SubscriptBox["a", "n"], "+", SubscriptBox["b", "n"]]], ")"]]]], "\[Equal]", RowBox[List[RowBox[List["ArithmeticGeometricMean", "[", RowBox[List[SubscriptBox["a", "0"], ",", SubscriptBox["b", "0"]]], "]"]], " ", SuperscriptBox[RowBox[List["EllipticTheta", "[", RowBox[List["3", ",", "0", ",", SuperscriptBox["z", SuperscriptBox["2", RowBox[List["n", "+", "1"]]]]]], "]"]], "2"]]]]], "&&", RowBox[List[SubscriptBox["b", RowBox[List["n", "+", "1"]]], "\[Equal]", SqrtBox[RowBox[List[SubscriptBox["a", "n"], " ", SubscriptBox["b", "n"]]]], "\[Equal]", RowBox[List[RowBox[List["ArithmeticGeometricMean", "[", RowBox[List[SubscriptBox["a", "0"], ",", SubscriptBox["b", "0"]]], "]"]], " ", SuperscriptBox[RowBox[List["EllipticTheta", "[", RowBox[List["4", ",", "0", ",", SuperscriptBox["z", SuperscriptBox["2", RowBox[List["n", "+", "1"]]]]]], "]"]], "2"]]]]], "&&", RowBox[List["z", "\[Equal]", RowBox[List["EllipticNomeQ", "[", RowBox[List["1", "-", SuperscriptBox[RowBox[List["(", FractionBox[SubscriptBox["b", "0"], SubscriptBox["a", "0"]], ")"]], "2"]]], "]"]]]]]]]]]]










Date Added to functions.wolfram.com (modification date)





2001-10-29





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