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EllipticF






Mathematica Notation

Traditional Notation









Elliptic Integrals > EllipticF[z,m] > Differentiation > Fractional integro-differentiation > With respect to z





http://functions.wolfram.com/08.05.20.0008.01









  


  










Input Form





D[EllipticF[z, m], {z, \[Alpha]}] == z^(1 - \[Alpha]) Sqrt[Pi] 2^\[Alpha] Sum[(-1)^k (Pochhammer[1/2, k]/k!) Hypergeometric2F1[k + 1/2, k + 1/2, 2 k + 1, m] HypergeometricPFQRegularized[{1}, {1 - \[Alpha]/2, (3 - \[Alpha])/2}, (-k^2) z^2] (m/4)^k, {k, 1, Infinity}] + (2 z^(1 - \[Alpha]) EllipticK[m])/ (Pi Gamma[2 - \[Alpha]])










Standard Form





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MathML Form







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</ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> EllipticK </ci> <ci> m </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["EllipticF", "[", RowBox[List["z_", ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List["1", "-", "\[Alpha]"]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["2", "\[Alpha]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", 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"]], "]"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]], ",", FractionBox[RowBox[List["3", "-", "\[Alpha]"]], "2"]]], "}"]], ",", RowBox[List[RowBox[List["-", SuperscriptBox["k", "2"]]], " ", SuperscriptBox["z", "2"]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["m", "4"], ")"]], "k"]]], RowBox[List["k", "!"]]]]]]], "+", FractionBox[RowBox[List["2", " ", SuperscriptBox["z", RowBox[List["1", "-", "\[Alpha]"]]], " ", RowBox[List["EllipticK", "[", "m", "]"]]]], RowBox[List["\[Pi]", " ", RowBox[List["Gamma", "[", RowBox[List["2", "-", "\[Alpha]"]], "]"]]]]]]]]]]]










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





2001-10-29