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ArithmeticGeometricMean




Mathematica Notation

Traditional Notation









Elliptic Functions > ArithmeticGeometricMean[a,b] > Representations through more general functions > Through hypergeometric functions > Involving 2F1





http://functions.wolfram.com/09.54.26.0001.01









  


  








Input Form





ArithmeticGeometricMean[a, b] == (a + b)/(2 HypergeometricPFQ[{1/2, 1/2}, {1}, ((a - b)/(a + b))^2])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["ArithmeticGeometricMean", "[", RowBox[List["a", ",", "b"]], "]"]], "\[Equal]", FractionBox[RowBox[List["a", "+", "b"]], RowBox[List["2", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox["1", "2"]]], "}"]], ",", RowBox[List["{", "1", "}"]], ",", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["a", "-", "b"]], RowBox[List["a", "+", "b"]]], ")"]], "2"]]], "]"]]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <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> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ; </mo> <mn> 1 </mn> <mo> ; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mrow> <mi> a </mi> <mo> + </mo> <mi> b </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, FormBox[&quot;2&quot;, TraditionalForm]], SubscriptBox[&quot;F&quot;, FormBox[&quot;1&quot;, TraditionalForm]]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox[&quot;1&quot;, &quot;2&quot;], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;1&quot;, &quot;2&quot;], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[TagBox[&quot;1&quot;, HypergeometricPFQ, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[SuperscriptBox[RowBox[List[&quot;(&quot;, FractionBox[RowBox[List[&quot;a&quot;, &quot;-&quot;, &quot;b&quot;]], RowBox[List[&quot;a&quot;, &quot;+&quot;, &quot;b&quot;]]], &quot;)&quot;]], &quot;2&quot;], HypergeometricPFQ, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> ArithmeticGeometricMean </ci> <ci> a </ci> <ci> b </ci> </apply> <apply> <times /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> HypergeometricPFQ </ci> <list> <cn type='rational'> 1 <sep /> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </list> <list> <cn type='integer'> 1 </cn> </list> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <ci> b </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArithmeticGeometricMean", "[", RowBox[List["a_", ",", "b_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["a", "+", "b"]], RowBox[List["2", " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", FractionBox["1", "2"]]], "}"]], ",", RowBox[List["{", "1", "}"]], ",", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["a", "-", "b"]], RowBox[List["a", "+", "b"]]], ")"]], "2"]]], "]"]]]]]]]]]


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





2001-10-29