Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











Cot






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.09.20.0007.01









  


  










Input Form





Derivative[\[Alpha]][Cot][c z] == Pi^(-\[Alpha] - 1) ((((-c) z)^\[Alpha] PolyGamma[\[Alpha], -((c z)/Pi)])/ (c z)^\[Alpha] - PolyGamma[\[Alpha], (c z)/Pi]) + Limit[((c z)^(-1 - \[Nu]) (EulerGamma + 2 Log[Pi] - Log[(-c) z] + PolyGamma[-\[Nu]]))/Gamma[-\[Nu]], \[Nu] -> \[Alpha]]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Derivative", "[", "\[Alpha]", "]"]], "[", "Cot", "]"]], "[", RowBox[List["c", " ", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["\[Pi]", RowBox[List[RowBox[List["-", "\[Alpha]"]], "-", "1"]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], " ", "z"]], ")"]], "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", "z"]], ")"]], RowBox[List["-", "\[Alpha]"]]], RowBox[List["PolyGamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List["-", FractionBox[RowBox[List[" ", RowBox[List["c", " ", "z"]]]], "\[Pi]"]]]]], "]"]]]], "-", RowBox[List["PolyGamma", "[", RowBox[List["\[Alpha]", ",", FractionBox[RowBox[List["c", " ", "z"]], "\[Pi]"]]], "]"]]]], ")"]]]], "+", RowBox[List["Limit", "[", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", "z"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List["EulerGamma", "+", RowBox[List["2", " ", RowBox[List["Log", "[", "\[Pi]", "]"]]]], "-", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "c"]], " ", "z"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]], ",", RowBox[List["\[Nu]", "\[Rule]", "\[Alpha]"]]]], "]"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mi> cot </mi> <semantics> <mrow> <mo> ( </mo> <mi> &#945; </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, &quot;\[Alpha]&quot;, &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mrow> <munder> <mi> lim </mi> <mrow> <mi> &#957; </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#945; </mi> </mrow> </munder> <mo> &#8290; </mo> <mtext> &#8201; </mtext> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mtext> </mtext> </mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#960; </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> c </mi> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mo> - </mo> <mi> &#957; </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> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#960; </mi> <mrow> <mrow> <mo> - </mo> <mi> &#945; </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> c </mi> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> &#945; </mi> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> &#945; </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mi> &#945; </mi> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mi> &#960; </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mi> &#945; </mi> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mi> &#960; </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> D </ci> <apply> <cot /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> <list> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> <ci> &#945; </ci> </list> </apply> <apply> <plus /> <apply> <limit /> <bvar> <ci> &#957; </ci> </bvar> <condition> <apply> <tendsto /> <ci> &#957; </ci> <ci> &#945; </ci> </apply> </condition> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ln /> <pi /> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <eulergamma /> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <pi /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> c </ci> </apply> <ci> z </ci> </apply> <ci> &#945; </ci> </apply> <apply> <power /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> <apply> <ci> PolyGamma </ci> <ci> &#945; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> c </ci> <ci> z </ci> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> PolyGamma </ci> <ci> &#945; </ci> <apply> <times /> <ci> c </ci> <ci> z </ci> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["Cot", TagBox[RowBox[List["(", "\[Alpha]", ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["c_", " ", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["\[Pi]", RowBox[List[RowBox[List["-", "\[Alpha]"]], "-", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], " ", "z"]], ")"]], "\[Alpha]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", "z"]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["PolyGamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List["-", FractionBox[RowBox[List["c", " ", "z"]], "\[Pi]"]]]]], "]"]]]], "-", RowBox[List["PolyGamma", "[", RowBox[List["\[Alpha]", ",", FractionBox[RowBox[List["c", " ", "z"]], "\[Pi]"]]], "]"]]]], ")"]]]], "+", RowBox[List["Limit", "[", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", "z"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List["EulerGamma", "+", RowBox[List["2", " ", RowBox[List["Log", "[", "\[Pi]", "]"]]]], "-", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", "c"]], " ", "z"]], "]"]], "+", RowBox[List["PolyGamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]], ",", RowBox[List["\[Nu]", "\[Rule]", "\[Alpha]"]]]], "]"]]]]]]]]










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





2007-05-02





© 1998-2014 Wolfram Research, Inc.