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EllipticF






Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticF[z,m] > Representations through more general functions > Through hypergeometric functions of two variables





http://functions.wolfram.com/08.05.26.0006.01









  


  










Input Form





EllipticF[z, m] == Sin[z] AppellF1[1/2, 1/2, 1/2, 3/2, Sin[z]^2, m Sin[z]^2] /; Abs[Re[z]] < Pi/2










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["EllipticF", "[", RowBox[List["z", ",", "m"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["Sin", "[", "z", "]"]], RowBox[List["AppellF1", "[", RowBox[List[FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox["3", "2"], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], ",", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Re", "[", "z", "]"]], "]"]], "<", FractionBox["\[Pi]", "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> <mrow> <mi> F </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> &#10072; </mo> <mi> m </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <msub> <mi> F </mi> <mn> 1 </mn> </msub> <annotation-xml encoding='MathML-Content'> <ci> AppellF1 </ci> </annotation-xml> </semantics> <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> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> ; </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> , </mo> <mrow> <mi> m </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sin </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> EllipticF </ci> <ci> z </ci> <ci> m </ci> </apply> <apply> <times /> <apply> <sin /> <ci> z </ci> </apply> <apply> <ci> AppellF1 </ci> <cn type='rational'> 1 <sep /> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> <cn type='rational'> 3 <sep /> 2 </cn> <apply> <power /> <apply> <sin /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <ci> m </ci> <apply> <power /> <apply> <sin /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <lt /> <apply> <abs /> <apply> <real /> <ci> z </ci> </apply> </apply> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["EllipticF", "[", RowBox[List["z_", ",", "m_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Sin", "[", "z", "]"]], " ", RowBox[List["AppellF1", "[", RowBox[List[FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox["1", "2"], ",", FractionBox["3", "2"], ",", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"], ",", RowBox[List["m", " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "2"]]]]], "]"]]]], "/;", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Re", "[", "z", "]"]], "]"]], "<", FractionBox["\[Pi]", "2"]]]]]]]]]










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





2001-10-29