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variants of this functions
Log






Mathematica Notation

Traditional Notation









Elementary Functions > Log[z] > Differentiation > Fractional integro-differentiation





http://functions.wolfram.com/01.04.20.0013.01









  


  










Input Form





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










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "\[Alpha]"]], "}"]]], RowBox[List["Log", "[", "z", "]"]]]], "\[Equal]", RowBox[List["Piecewise", "[", RowBox[List[RowBox[List["{", RowBox[List["{", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["\[Alpha]", "-", "1"]], ")"]], "!"]], SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]]]], ",", RowBox[List[RowBox[List["Element", "[", RowBox[List["\[Alpha]", ",", "Integers"]], "]"]], "\[And]", RowBox[List["\[Alpha]", ">", "0"]]]]]], "}"]], "}"]], ",", RowBox[List[FractionBox[RowBox[List[RowBox[List["Log", "[", "z", "]"]], "-", "EulerGamma", "-", RowBox[List["PolyGamma", "[", RowBox[List["1", "-", "\[Alpha]"]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Alpha]"]], "]"]]], SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]]]]]], "]"]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mi> &#945; </mi> </msup> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> z </mi> <mi> &#945; </mi> </msup> </mrow> </mfrac> <mo> &#63449; </mo> <mrow> <mo> &#62305; </mo> <mtable> <mtr> <mtd> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> &#945; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> &#945; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mi> &#945; </mi> </mrow> </msup> </mrow> </mtd> <mtd> <mrow> <mi> &#945; </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#945; </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[List[], EulerGamma]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mi> &#945; </mi> </mrow> </msup> </mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#945; </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> </mtd> <mtd> <semantics> <mi> True </mi> <annotation encoding='Mathematica'> TagBox[&quot;True&quot;, &quot;PiecewiseDefault&quot;, Rule[AutoDelete, False], Rule[DeletionWarning, True]] </annotation> </semantics> </mtd> </mtr> </mtable> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <ci> &#945; </ci> </degree> </bvar> <apply> <ln /> <ci> z </ci> </apply> </apply> <piecewise> <piece> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <factorial /> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> </apply> <apply> <in /> <ci> &#945; </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </piece> <otherwise> <apply> <times /> <apply> <plus /> <apply> <ln /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <eulergamma /> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </otherwise> </piecewise> </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["Log", "[", "z_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List["\[Piecewise]", GridBox[List[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Alpha]", "-", "1"]]], " ", RowBox[List[RowBox[List["(", RowBox[List["\[Alpha]", "-", "1"]], ")"]], "!"]], " ", SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]]]], RowBox[List[RowBox[List["\[Alpha]", "\[Element]", "Integers"]], "&&", RowBox[List["\[Alpha]", ">", "0"]]]]], List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "z", "]"]], "-", "EulerGamma", "-", RowBox[List["PolyGamma", "[", RowBox[List["1", "-", "\[Alpha]"]], "]"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[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