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variants of this functions
PolyGamma




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > PolyGamma[nu,z] > Continued fraction representations





http://functions.wolfram.com/06.15.10.0005.01









  


  








Input Form





PolyGamma[2, z] == -(1/z^3) - 1/z^2 - (1/(2 z^3)) (1/ (z + 1/(3 (z + 2/(3 (z + 6/(5 (z + 9/(5 (z + 18/(7 (z + 24/(7 (z + 40/(9 (z + \[Ellipsis])))))))))))))))) /; Re[z] > 0








Standard Form





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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> <mn> 2 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mfrac> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mfrac> <mn> 1 </mn> <mrow> <mi> z </mi> <mo> + </mo> <mstyle scriptlevel='0'> <mfrac> <mrow> <mn> 1 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> <mstyle scriptlevel='0'> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> <mstyle scriptlevel='0'> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 6 </mn> <mo> / </mo> <mn> 5 </mn> </mrow> <mstyle scriptlevel='0'> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 9 </mn> <mo> / </mo> <mn> 5 </mn> </mrow> <mstyle scriptlevel='0'> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 18 </mn> <mo> / </mo> <mn> 7 </mn> </mrow> <mstyle scriptlevel='0'> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 24 </mn> <mo> / </mo> <mn> 7 </mn> </mrow> <mstyle scriptlevel='0'> <mrow> <mi> z </mi> <mo> + </mo> <mfrac> <mrow> <mn> 40 </mn> <mo> / </mo> <mn> 9 </mn> </mrow> <mstyle scriptlevel='0'> <mrow> <mi> z </mi> <mo> + </mo> <mo> &#8230; </mo> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mrow> </mstyle> </mfrac> </mstyle> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> PolyGamma </ci> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <power /> <cn type='integer'> 5 </cn> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 18 </cn> <apply> <power /> <cn type='integer'> 7 </cn> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 24 </cn> <apply> <power /> <cn type='integer'> 7 </cn> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> 40 </cn> <apply> <power /> <cn type='integer'> 9 </cn> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <ci> &#8230; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <gt /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyGamma", "[", RowBox[List["2", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", SuperscriptBox["z", "3"]]]], "-", FractionBox["1", SuperscriptBox["z", "2"]], "-", FractionBox["1", RowBox[List[RowBox[List["(", RowBox[List["2", " ", SuperscriptBox["z", "3"]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "+", FractionBox["1", RowBox[List["3", " ", RowBox[List["(", RowBox[List["z", "+", FractionBox["2", RowBox[List["3", " ", RowBox[List["(", RowBox[List["z", "+", FractionBox["6", RowBox[List["5", " ", RowBox[List["(", RowBox[List["z", "+", FractionBox["9", RowBox[List["5", " ", RowBox[List["(", RowBox[List["z", "+", FractionBox["18", RowBox[List["7", " ", RowBox[List["(", RowBox[List["z", "+", FractionBox["24", RowBox[List["7", " ", RowBox[List["(", RowBox[List["z", "+", FractionBox["40", RowBox[List["9", " ", RowBox[List["(", RowBox[List["z", "+", "\[Ellipsis]"]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]], ")"]]]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]]]]]


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





2001-10-29