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
 











Tan






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.08.20.0007.01









  


  










Input Form





Derivative[\[Alpha]][Tan][c z] == (Log[4] (c z)^(-1 - \[Alpha]))/Gamma[-\[Alpha]] - (Pi^(-1 - \[Alpha]) (((-c) z)^\[Alpha] (2^(1 + \[Alpha]) PolyGamma[\[Alpha], -((2 c z)/Pi)] - PolyGamma[\[Alpha], -((c z)/Pi)]) + (c z)^\[Alpha] (PolyGamma[\[Alpha], (c z)/Pi] - 2^(1 + \[Alpha]) PolyGamma[\[Alpha], (2 c z)/Pi])))/(c z)^\[Alpha]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["Derivative", "[", "\[Alpha]", "]"]], "[", "Tan", "]"]], "[", RowBox[List["c", " ", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Log", "[", "4", "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", "z"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "\[Alpha]"]]]]], RowBox[List["Gamma", "[", RowBox[List["-", "\[Alpha]"]], "]"]]], "-", RowBox[List[SuperscriptBox["\[Pi]", RowBox[List[RowBox[List["-", "1"]], "-", "\[Alpha]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", "z"]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], " ", "z"]], ")"]], "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["1", "+", "\[Alpha]"]]], " ", RowBox[List["PolyGamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List["-", FractionBox[RowBox[List["2", " ", "c", " ", "z"]], "\[Pi]"]]]]], "]"]]]], "-", RowBox[List["PolyGamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List["-", FractionBox[RowBox[List["c", " ", "z"]], "\[Pi]"]]]]], "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", "z"]], ")"]], "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["\[Alpha]", ",", FractionBox[RowBox[List["c", " ", "z"]], "\[Pi]"]]], "]"]], "-", RowBox[List[SuperscriptBox["2", RowBox[List["1", "+", "\[Alpha]"]]], " ", RowBox[List["PolyGamma", "[", RowBox[List["\[Alpha]", ",", FractionBox[RowBox[List["2", " ", "c", " ", "z"]], "\[Pi]"]]], "]"]]]]]], ")"]]]]]], ")"]]]]]]]]]]










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> tan </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> <mfrac> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 4 </mn> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> &#945; </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> &#945; </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <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> <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> <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> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mn> 2 </mn> <mrow> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </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> <mn> 2 </mn> <mo> &#8290; </mo> <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> <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> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> &#945; </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <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> <mo> - </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mi> &#945; </mi> <mo> + </mo> <mn> 1 </mn> </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> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mi> &#960; </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> D </ci> <apply> <tan /> <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> <times /> <apply> <ln /> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <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> <power /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#945; </ci> </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> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> PolyGamma </ci> <ci> &#945; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <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 /> <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> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> <ci> &#945; </ci> </apply> <apply> <plus /> <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> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> &#945; </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> PolyGamma </ci> <ci> &#945; </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> c </ci> <ci> z </ci> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SuperscriptBox["Tan", TagBox[RowBox[List["(", "\[Alpha]", ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["c_", " ", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["Log", "[", "4", "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", "z"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "\[Alpha]"]]]]], RowBox[List["Gamma", "[", RowBox[List["-", "\[Alpha]"]], "]"]]], "-", RowBox[List[SuperscriptBox["\[Pi]", RowBox[List[RowBox[List["-", "1"]], "-", "\[Alpha]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", "z"]], ")"]], RowBox[List["-", "\[Alpha]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "c"]], " ", "z"]], ")"]], "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["1", "+", "\[Alpha]"]]], " ", RowBox[List["PolyGamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List["-", FractionBox[RowBox[List["2", " ", "c", " ", "z"]], "\[Pi]"]]]]], "]"]]]], "-", RowBox[List["PolyGamma", "[", RowBox[List["\[Alpha]", ",", RowBox[List["-", FractionBox[RowBox[List["c", " ", "z"]], "\[Pi]"]]]]], "]"]]]], ")"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", "z"]], ")"]], "\[Alpha]"], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["\[Alpha]", ",", FractionBox[RowBox[List["c", " ", "z"]], "\[Pi]"]]], "]"]], "-", RowBox[List[SuperscriptBox["2", RowBox[List["1", "+", "\[Alpha]"]]], " ", RowBox[List["PolyGamma", "[", RowBox[List["\[Alpha]", ",", FractionBox[RowBox[List["2", " ", "c", " ", "z"]], "\[Pi]"]]], "]"]]]]]], ")"]]]]]], ")"]]]]]]]]]]










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





2007-05-02





© 1998-2014 Wolfram Research, Inc.