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InverseJacobiDN






Mathematica Notation

Traditional Notation









Elliptic Functions > InverseJacobiDN[z,m] > Differentiation > Fractional integro-differentiation > With respect to m





http://functions.wolfram.com/09.41.20.0010.01









  


  










Input Form





D[InverseJacobiDN[z, m], {m, \[Alpha]}] == I Sum[(1/(k! Gamma[k - \[Alpha] + 1])) Pochhammer[1/2, k]^2 (PolyGamma[k + 1] - PolyGamma[1/2 + k]) m^k, {k, 0, Infinity}] - (I/(m^\[Alpha] 2)) Sum[(Pochhammer[1/2, k]^2 FDLogConstant[m, k, \[Alpha]] m^k)/k!^2, {k, 0, Infinity}] - (Sqrt[Pi]/(m^\[Alpha] (2 Sqrt[1 - z^2]))) HypergeometricPFQRegularized[ {{1/2}, {1/2}, {1/2, 1}}, {{3/2}, {}, {1 - \[Alpha]}}, 1/(1 - z^2), m/(1 - z^2)] /; z > 1 && -1 < m < 0










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["m_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["InverseJacobiDN", "[", RowBox[List["z_", ",", "m_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["k", "+", "1"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k"]], "]"]]]], ")"]], " ", SuperscriptBox["m", "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["k", "-", "\[Alpha]", "+", "1"]], "]"]]]]]]]]], "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["m", RowBox[List["-", "\[Alpha]"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]], "2"], " ", RowBox[List["FDLogConstant", "[", RowBox[List["m", ",", "k", ",", "\[Alpha]"]], "]"]], " ", SuperscriptBox["m", "k"]]], SuperscriptBox[RowBox[List["(", RowBox[List["k", "!"]], ")"]], "2"]]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["m", RowBox[List["-", "\[Alpha]"]]], " ", SqrtBox["\[Pi]"]]], ")"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox["1", "2"], "}"]], ",", RowBox[List["{", FractionBox["1", "2"], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", "1"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox["3", "2"], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["1", "-", "\[Alpha]"]], "}"]]]], "}"]], ",", FractionBox["1", RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], ",", FractionBox["m", RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]], "]"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]]]], "/;", RowBox[List[RowBox[List["z", ">", "1"]], "&&", RowBox[List[RowBox[List["-", "1"]], "<", "m", "<", "0"]]]]]]]]]]










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





2001-10-29





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