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variants of this functions
PolyGamma






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > PolyGamma[z] > Series representations > Generalized power series > Expansions at z==-n > For the function itself





http://functions.wolfram.com/06.14.06.0019.01









  


  










Input Form





PolyGamma[z] \[Proportional] -(1/(z + n)) + PolyGamma[n + 1] + (Pi^2/3 - Zeta[2, 1 + n]) (z + n) - Zeta[3, 1 + n] (z + n)^2 + (Pi^4/45 - Zeta[4, 1 + n]) (z + n)^3 + O[(z + n)^4] /; Element[n, Integers] && n >= 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["PolyGamma", "[", "z", "]"]], "\[Proportional]", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["z", "+", "n"]]]]], "+", RowBox[List["PolyGamma", "[", RowBox[List["n", "+", "1"]], "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[FractionBox[SuperscriptBox["\[Pi]", "2"], "3"], "-", RowBox[List["Zeta", "[", RowBox[List["2", ",", RowBox[List["1", "+", "n"]]]], "]"]]]], ")"]], RowBox[List["(", RowBox[List["z", "+", "n"]], ")"]]]], "-", " ", RowBox[List[RowBox[List["Zeta", "[", RowBox[List["3", ",", RowBox[List["1", "+", "n"]]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "n"]], ")"]], "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[FractionBox[SuperscriptBox["\[Pi]", "4"], "45"], "-", RowBox[List["Zeta", "[", RowBox[List["4", ",", RowBox[List["1", "+", "n"]]]], "]"]]]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "n"]], ")"]], "3"]]], " ", "+", RowBox[List["O", "[", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "n"]], ")"]], "4"], "]"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "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> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mi> z </mi> <mo> + </mo> <mi> n </mi> </mrow> </mfrac> </mrow> <mo> + </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mn> 3 </mn> </mfrac> <mo> - </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> , </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, RowBox[List[TagBox[&quot;2&quot;, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;n&quot;, &quot;+&quot;, &quot;1&quot;]], Rule[Editable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2], Zeta[$CellContext`e1, $CellContext`e2]]]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> , </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, RowBox[List[TagBox[&quot;3&quot;, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;n&quot;, &quot;+&quot;, &quot;1&quot;]], Rule[Editable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2], Zeta[$CellContext`e1, $CellContext`e2]]]] </annotation> </semantics> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <msup> <mi> &#960; </mi> <mn> 4 </mn> </msup> <mn> 45 </mn> </mfrac> <mo> - </mo> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 4 </mn> <mo> , </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, RowBox[List[TagBox[&quot;4&quot;, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;n&quot;, &quot;+&quot;, &quot;1&quot;]], Rule[Editable, True]]]], &quot;)&quot;]], InterpretTemplate[Function[List[$CellContext`e1, $CellContext`e2], Zeta[$CellContext`e1, $CellContext`e2]]]] </annotation> </semantics> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> PolyGamma </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <ci> z </ci> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Zeta </ci> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <ci> z </ci> <ci> n </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Zeta </ci> <cn type='integer'> 3 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <ci> n </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <cn type='integer'> 45 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Zeta </ci> <cn type='integer'> 4 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <ci> n </ci> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <apply> <plus /> <ci> z </ci> <ci> n </ci> </apply> <cn type='integer'> 4 </cn> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["PolyGamma", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["z", "+", "n"]]]]], "+", RowBox[List["PolyGamma", "[", RowBox[List["n", "+", "1"]], "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[FractionBox[SuperscriptBox["\[Pi]", "2"], "3"], "-", RowBox[List["Zeta", "[", RowBox[List["2", ",", RowBox[List["1", "+", "n"]]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "+", "n"]], ")"]]]], "-", RowBox[List[RowBox[List["Zeta", "[", RowBox[List["3", ",", RowBox[List["1", "+", "n"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "n"]], ")"]], "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[FractionBox[SuperscriptBox["\[Pi]", "4"], "45"], "-", RowBox[List["Zeta", "[", RowBox[List["4", ",", RowBox[List["1", "+", "n"]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "n"]], ")"]], "3"]]], "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["z", "+", "n"]], "]"]], "4"]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2007-05-02





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