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]

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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> <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>

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2001-10-29