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variants of this functions
ArcTan






Mathematica Notation

Traditional Notation









Elementary Functions > ArcTan[x,y] > Differentiation > Fractional integro-differentiation > With respect to y





http://functions.wolfram.com/01.15.20.0008.01









  


  










Input Form





D[ArcTan[x, y], {y, \[Alpha]}] == (I/2) (1/(Sqrt[1/x^2] x) - 1) ((Log[-y] - Log[y])/ (y^\[Alpha] Gamma[1 - \[Alpha]])) + ((2^(-1 + \[Alpha]) Sqrt[Pi] y^(1 - \[Alpha]))/x) HypergeometricPFQRegularized[{1/2, 1, 1}, {1 - \[Alpha]/2, 3/2 - \[Alpha]/2}, -(y^2/x^2)] /; Abs[y] < Abs[x] && Element[y, Reals]










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["y_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["ArcTan", "[", RowBox[List["x_", ",", "y_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[FractionBox["1", RowBox[List[SqrtBox[FractionBox["1", SuperscriptBox["x", "2"]]], " ", "x"]]], "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Log", "[", RowBox[List["-", "y"]], "]"]], "-", RowBox[List["Log", "[", "y", "]"]]]], ")"]], " ", SuperscriptBox["y", RowBox[List["-", "\[Alpha]"]]]]], ")"]]]], RowBox[List["2", " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Alpha]"]], "]"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", "\[Alpha]"]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["y", RowBox[List["1", "-", "\[Alpha]"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", "1", ",", "1"]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", FractionBox["\[Alpha]", "2"]]], ",", RowBox[List[FractionBox["3", "2"], "-", FractionBox["\[Alpha]", "2"]]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["y", "2"], SuperscriptBox["x", "2"]]]]]], "]"]]]], "x"]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "y", "]"]], "<", RowBox[List["Abs", "[", "x", "]"]]]], "&&", RowBox[List["y", "\[Element]", "Reals"]]]]]]]]]]










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





2001-10-29