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variants of this functions
EllipticE






Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticE[z] > Identities > Functional identities





http://functions.wolfram.com/08.01.17.0006.01









  


  










Input Form





EllipticE[z] == Sqrt[z] EllipticE[1/z] - Sqrt[-(1/z)] Sqrt[1/(1 - z)] Sqrt[1 - z] Sqrt[z] EllipticE[1 - z] + (1 - z) EllipticK[z] + z Sqrt[-(1/z)] Sqrt[1/(1 - z)] Sqrt[z (1 - z)] EllipticK[1 - z]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["EllipticE", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SqrtBox["z"], " ", RowBox[List["EllipticE", "[", FractionBox["1", "z"], "]"]]]], "-", RowBox[List[SqrtBox[RowBox[List["-", FractionBox["1", "z"]]]], SqrtBox[FractionBox["1", RowBox[List["1", "-", "z"]]]], " ", SqrtBox[RowBox[List["1", "-", "z"]]], SqrtBox["z"], " ", RowBox[List["EllipticE", "[", RowBox[List["1", "-", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], " ", RowBox[List["EllipticK", "[", "z", "]"]]]], "+", RowBox[List["z", SqrtBox[RowBox[List["-", FractionBox["1", "z"]]]], SqrtBox[FractionBox["1", RowBox[List["1", "-", "z"]]]], " ", SqrtBox[RowBox[List["z", RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]]]]], RowBox[List["EllipticK", "[", RowBox[List["1", "-", "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> &#63449; </mo> <mrow> <mrow> <msqrt> <mi> z </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mi> E </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msqrt> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mi> E </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> EllipticE </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> EllipticE </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> EllipticE </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> <apply> <ci> EllipticK </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <ci> EllipticK </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticE", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SqrtBox["z"], " ", RowBox[List["EllipticE", "[", FractionBox["1", "z"], "]"]]]], "-", RowBox[List[SqrtBox[RowBox[List["-", FractionBox["1", "z"]]]], " ", SqrtBox[FractionBox["1", RowBox[List["1", "-", "z"]]]], " ", SqrtBox[RowBox[List["1", "-", "z"]]], " ", SqrtBox["z"], " ", RowBox[List["EllipticE", "[", RowBox[List["1", "-", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], " ", RowBox[List["EllipticK", "[", "z", "]"]]]], "+", RowBox[List["z", " ", SqrtBox[RowBox[List["-", FractionBox["1", "z"]]]], " ", SqrtBox[FractionBox["1", RowBox[List["1", "-", "z"]]]], " ", SqrtBox[RowBox[List["z", " ", RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]]]]], " ", RowBox[List["EllipticK", "[", RowBox[List["1", "-", "z"]], "]"]]]]]]]]]]










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





2007-05-02