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EllipticF






Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticF[z,m] > Integration > Indefinite integration > Involving only one direct function





http://functions.wolfram.com/08.05.21.0001.01









  


  










Input Form





Integrate[EllipticF[z, m], z] == (z^2 EllipticK[m])/Pi + (1/2) Sum[(((-1)^k (1 - Cos[2 k z]))/(k! k^2)) Pochhammer[1/2, k] Hypergeometric2F1[k + 1/2, k + 1/2, 2 k + 1, m] (m/4)^k, {k, 1, Infinity}]










Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> &#8747; </mo> <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> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mi> &#960; </mi> </mfrac> <mo> &#8290; </mo> <mrow> <mi> K </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> m </mi> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> k </mi> <mo> ! 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List["EllipticF", "[", RowBox[List["z_", ",", "m_"]], "]"]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], "\[Pi]"], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Cos", "[", RowBox[List["2", " ", "k", " ", "z"]], "]"]]]], ")"]]]], ")"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], " ", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["k", "+", FractionBox["1", "2"]]], ",", RowBox[List["k", "+", FractionBox["1", "2"]]], ",", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ",", "m"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["m", "4"], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", SuperscriptBox["k", "2"]]]]]]]]]]]]]]










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





2001-10-29





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