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Power






Mathematica Notation

Traditional Notation









Elementary Functions > Power[z,a] > Differentiation > Fractional integro-differentiation > With respect to z





http://functions.wolfram.com/01.02.20.0026.01









  


  










Input Form





D[z^a, {z, \[Alpha]}] == Piecewise[{{(-1)^\[Alpha] Pochhammer[-a, \[Alpha]] z^(a - \[Alpha]), Element[\[Alpha], Integers] && Element[a, Integers] && a < 0 && a < \[Alpha]}, {(((-1)^(a - 1) (Log[z] - PolyGamma[1 + a - \[Alpha]] + PolyGamma[-a]))/ ((-a - 1)! Gamma[1 + a - \[Alpha]])) z^(a - \[Alpha]), Element[a, Integers] && a < 0}}, (Gamma[1 + a]/Gamma[1 + a - \[Alpha]]) z^(a - \[Alpha])]










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]_"]], "}"]]]]], SuperscriptBox["z_", "a_"]]], "]"]], "\[RuleDelayed]", RowBox[List["\[Piecewise]", GridBox[List[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "\[Alpha]"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "a"]], ",", "\[Alpha]"]], "]"]], " ", SuperscriptBox["z", RowBox[List["a", "-", "\[Alpha]"]]]]], RowBox[List[RowBox[List["\[Alpha]", "\[Element]", "Integers"]], "&&", RowBox[List["a", "\[Element]", "Integers"]], "&&", RowBox[List["a", "<", "0"]], "&&", RowBox[List["a", "<", "\[Alpha]"]]]]], List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["a", "-", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "z", "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "a", "-", "\[Alpha]"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["-", "a"]], "]"]]]], ")"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["a", "-", "\[Alpha]"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], "-", "1"]], ")"]], "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "\[Alpha]"]], "]"]]]]], RowBox[List[RowBox[List["a", "\[Element]", "Integers"]], "&&", RowBox[List["a", "<", "0"]]]]], List[FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "+", "a"]], "]"]], " ", SuperscriptBox["z", RowBox[List["a", "-", "\[Alpha]"]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "a", "-", "\[Alpha]"]], "]"]]], TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]]]], Rule[ColumnAlignments, List[Left]], Rule[ColumnSpacings, 1.2`], Rule[ColumnWidths, Automatic]]]]]]]]










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





2007-05-02





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