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variants of this functions
EllipticE




Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticE[z] > Representations through more general functions > Through other functions > Involving Legendre functions





http://functions.wolfram.com/08.01.26.0074.01









  


  








Input Form





EllipticE[z] == (Pi/4) (LegendreP[-(1/2), 1 - 2 z] + LegendreP[1/2, 1 - 2 z])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["EllipticE", "[", "z", "]"]], "\[Equal]", RowBox[List[FractionBox["\[Pi]", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["LegendreP", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["1", "-", RowBox[List["2", " ", "z"]]]]]], "]"]], "+", RowBox[List["LegendreP", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["1", "-", RowBox[List["2", " ", "z"]]]]]], "]"]]]], ")"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> E </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mi> &#960; </mi> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </msub> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <msub> <semantics> <mi> P </mi> <annotation encoding='Mathematica'> TagBox[&quot;P&quot;, LegendreP] </annotation> </semantics> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msub> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> EllipticE </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> LegendreP </ci> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <ci> LegendreP </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticE", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["LegendreP", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["1", "-", RowBox[List["2", " ", "z"]]]]]], "]"]], "+", RowBox[List["LegendreP", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["1", "-", RowBox[List["2", " ", "z"]]]]]], "]"]]]], ")"]]]]]]]]


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





2001-10-29