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
 











variants of this functions
PolyGamma






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > PolyGamma[nu,z] > Series representations > Generalized power series > Expansions at z==0 > For the function itself > General case





http://functions.wolfram.com/06.15.06.0023.01









  


  










Input Form





PolyGamma[\[Nu], z] \[Proportional] (-FDPowerConstant[z, -1, \[Nu]]) z^(-1 - \[Nu]) - EulerGamma/(z^\[Nu] Gamma[1 - \[Nu]]) + ((Pi^2 z^(1 - \[Nu]))/(6 Gamma[2 - \[Nu]])) (1 - (12 z Zeta[3])/(Pi^2 (2 - \[Nu])) + (2 Pi^2 z^2)/ (5 (2 - \[Nu]) (3 - \[Nu])) + \[Ellipsis]) /; (z -> 0) && Re[\[Nu]] > 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["FDPowerConstant", "[", RowBox[List["z", ",", RowBox[List["-", "1"]], ",", "\[Nu]"]], "]"]]]], SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]]]], "-", FractionBox[RowBox[List["EulerGamma", " ", SuperscriptBox["z", RowBox[List["-", "\[Nu]"]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", RowBox[List["1", "-", "\[Nu]"]]]]], RowBox[List["6", " ", RowBox[List["Gamma", "[", RowBox[List["2", "-", "\[Nu]"]], "]"]]]]], RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List["12", " ", "z", " ", RowBox[List["Zeta", "[", "3", "]"]]]], RowBox[List[SuperscriptBox["\[Pi]", "2"], RowBox[List["(", RowBox[List["2", "-", "\[Nu]"]], ")"]]]]], "+", FractionBox[RowBox[List["2", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", "2"]]], RowBox[List["5", " ", RowBox[List["(", RowBox[List["2", "-", "\[Nu]"]], ")"]], RowBox[List["(", RowBox[List["3", "-", "\[Nu]"]], ")"]]]]], "+", "\[Ellipsis]"]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", "0"]], ")"]], "\[And]", RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], ">", "0"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mtext> </mtext> <mrow> <msubsup> <mi> &#8497;&#119966; </mi> <mi> exp </mi> <mrow> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> </msubsup> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mo> - </mo> <mfrac> <mrow> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[EulerGamma]] </annotation> </semantics> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> </msup> </mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> + </mo> <mrow> <mfrac> <mrow> <mtext> </mtext> <mrow> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#957; </mi> </mrow> </msup> </mrow> </mrow> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <mn> 12 </mn> <mo> &#8290; </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[&quot;3&quot;, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mrow> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <mn> 5 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> PolyGamma </ci> <ci> &#957; </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> &#8497;&#119966; </ci> <ci> exp </ci> </apply> <ci> &#957; </ci> </apply> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <eulergamma /> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 12 </cn> <apply> <ci> Zeta </ci> <cn type='integer'> 3 </cn> </apply> <ci> z </ci> <apply> <power /> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 5 </cn> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <plus /> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <gt /> <apply> <real /> <ci> &#957; </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyGamma", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["FDPowerConstant", "[", RowBox[List["z", ",", RowBox[List["-", "1"]], ",", "\[Nu]"]], "]"]]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]]]], "-", FractionBox[RowBox[List["EulerGamma", " ", SuperscriptBox["z", RowBox[List["-", "\[Nu]"]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", RowBox[List["1", "-", "\[Nu]"]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List["12", " ", "z", " ", RowBox[List["Zeta", "[", "3", "]"]]]], RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", RowBox[List["(", RowBox[List["2", "-", "\[Nu]"]], ")"]]]]], "+", FractionBox[RowBox[List["2", " ", SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", "2"]]], RowBox[List["5", " ", RowBox[List["(", RowBox[List["2", "-", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "-", "\[Nu]"]], ")"]]]]], "+", "\[Ellipsis]"]], ")"]]]], RowBox[List["6", " ", RowBox[List["Gamma", "[", RowBox[List["2", "-", "\[Nu]"]], "]"]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", "0"]], ")"]], "&&", RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], ">", "0"]]]]]]]]]]










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





2007-05-02





© 1998-2014 Wolfram Research, Inc.