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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/08.05.20.0009.01









  


  










Input Form





D[EllipticF[z, m], {z, \[Alpha]}] == Sqrt[Pi] z^(1 - \[Alpha]) 2^(\[Alpha] - 1) Sum[(((-1)^(j + l) 2^(-2 j - 2 l) m^l)/ ((1 + 2 j + 2 l) j! l!)) Pochhammer[1/2, j] Pochhammer[1/2, l] Sum[(-1)^p Binomial[1 + 2 j + 2 l, p] (1 + 2 j + 2 l - 2 p) HypergeometricPFQRegularized[{1}, {1 - \[Alpha]/2, (3 - \[Alpha])/2}, -(((1 + 2 j + 2 l - 2 p)^2 z^2)/4)], {p, 0, j + l}], {j, 0, Infinity}, {l, 0, Infinity}]










Standard Form





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MathML Form







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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[SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List["1", "-", "\[Alpha]"]]], " ", SuperscriptBox["2", RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["l", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["j", "+", "l"]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "j"]], "-", RowBox[List["2", " ", "l"]]]]], " ", SuperscriptBox["m", "l"]]], ")"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "l"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "0"]], RowBox[List["j", "+", "l"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "p"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["1", "+", RowBox[List["2", " ", "j"]], "+", RowBox[List["2", " ", "l"]]]], ",", "p"]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "j"]], "+", RowBox[List["2", " ", "l"]], "-", RowBox[List["2", " ", "p"]]]], ")"]], " ", 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["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "j"]], "+", RowBox[List["2", " ", "l"]], "-", RowBox[List["2", " ", "p"]]]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]], ")"]]]]]], "]"]]]]]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "j"]], "+", RowBox[List["2", " ", "l"]]]], ")"]], " ", RowBox[List["j", "!"]], " ", RowBox[List["l", "!"]]]]]]]]]]]]]]]










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





2001-10-29





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