Polygamma function: Primary definition (formula 06.15.02.0007)

PolyGamma

$Mathematica Notation$

Gamma, Beta, Erf

PolyGamma[nu,z]

Primary definition

http://functions.wolfram.com/06.15.02.0007.01

Input Form

PolyGamma[\[Nu], z] == Piecewise[{{-(1/z) - EulerGamma + z Sum[1/(k (k + z)), {k, 1, Infinity}], \[Nu] == 0}, {(-1)^(\[Nu] + 1) \[Nu]! Sum[1/(k + z)^(\[Nu] + 1), {k, 0, Infinity}], Element[\[Nu], Integers] && \[Nu] > 0}, {((\[Nu] Log[z] - EulerGamma (z + \[Nu]) - \[Nu] PolyGamma[-\[Nu]])/ Gamma[1 - \[Nu]]) z^(-1 - \[Nu]) + z^(1 - \[Nu]) Sum[(1/k^2) Hypergeometric2F1Regularized[1, 2, 2 - \[Nu], -(z/k)], {k, 1, Infinity}], True}}]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List["Piecewise", "[", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "z"]]], "-", "EulerGamma", " ", "+", RowBox[List["z", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox["1", RowBox[List["k", RowBox[List["(", RowBox[List["k", "+", "z"]], ")"]]]]]]]]]]], ",", RowBox[List["\[Nu]", "\[Equal]", "0"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "+", "1"]]], " ", RowBox[List["\[Nu]", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["k", "+", "z"]], ")"]], RowBox[List["\[Nu]", "+", "1"]]]]]]]], ",", RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "\[And]", RowBox[List["\[Nu]", ">", "0"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["\[Nu]", " ", RowBox[List["Log", "[", "z", "]"]]]], "-", RowBox[List["EulerGamma", " ", RowBox[List["(", RowBox[List["z", "+", "\[Nu]"]], ")"]]]], "-", RowBox[List["\[Nu]", " ", RowBox[List["PolyGamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]]], SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]]]], " ", "+", RowBox[List[SuperscriptBox["z", RowBox[List["1", "-", "\[Nu]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List[FractionBox["1", SuperscriptBox["k", "2"]], " ", RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["1", ",", "2", ",", RowBox[List["2", "-", "\[Nu]"]], ",", RowBox[List["-", FractionBox["z", "k"]]]]], "]"]]]]]]]]]], ",", "True"]], "}"]]]], "}"]], "]"]]]]]]

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> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <mo>  </mo> <mtable> <mtr> <mtd> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </mrow> <mo> - </mo> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] </annotation> </semantics> <mo> + </mo> <mrow> <mi> z </mi> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mn> 1 </mn> <mrow> <mi> k </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> </mrow> </mtd> <mtd> <mrow> <mi> ν </mi> <mo>  </mo> <mn> 0 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> ν </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mn> 1 </mn> <msup> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> </mfrac> </mrow> </mrow> </mtd> <mtd> <mrow> <mi> ν </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mfrac> <mrow> <mrow> <mrow> <mo> - </mo> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] </annotation> </semantics> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> ν </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> ν </mi> <mo> ⁢ </mo> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <msup> <mi> z </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> ν </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mfrac> <mn> 1 </mn> <msup> <mi> k </mi> <mn> 2 </mn> </msup> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> </mrow> <mo> ; </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <mi> z </mi> <mi> k </mi> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox[OverscriptBox["F", "~"], "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", Hypergeometric2F1Regularized, Rule[Editable, True]], ",", TagBox["2", Hypergeometric2F1Regularized, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["2", "-", "\[Nu]"]], Hypergeometric2F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric2F1Regularized, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox["z", "k"]]], Hypergeometric2F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric2F1Regularized] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> </mtd> <mtd> <mi> True </mi> </mtd> </mtr> </mtable> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> PolyGamma </ci> <ci> ν </ci> <ci> z </ci> </apply> <apply> <ci> Piecewise </ci> <list> <list> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <eulergamma /> </apply> <apply> <times /> <ci> z </ci> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <ci> k </ci> <apply> <plus /> <ci> k </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </list> <list> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <factorial /> <ci> ν </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> k </ci> <ci> z </ci> </apply> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <in /> <ci> ν </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </list> <list> <apply> <plus /> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <eulergamma /> </apply> <apply> <plus /> <ci> z </ci> <ci> ν </ci> </apply> </apply> <apply> <times /> <ci> ν </ci> <apply> <ln /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> ν </ci> <apply> <ci> PolyGamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Hypergeometric2F1Regularized </ci> <cn type='integer'> 1 </cn> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <true /> </list> </list> </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["\[Piecewise]", GridBox[List[List[RowBox[List[RowBox[List["-", FractionBox["1", "z"]]], "-", "EulerGamma", "+", RowBox[List["z", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox["1", RowBox[List["k", " ", RowBox[List["(", RowBox[List["k", "+", "z"]], ")"]]]]]]]]]]], RowBox[List["\[Nu]", "\[Equal]", "0"]]], List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["\[Nu]", "+", "1"]]], " ", RowBox[List["\[Nu]", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["k", "+", "z"]], ")"]], RowBox[List["\[Nu]", "+", "1"]]]]]]]], RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "&&", RowBox[List["\[Nu]", ">", "0"]]]]], List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["\[Nu]", " ", RowBox[List["Log", "[", "z", "]"]]]], "-", RowBox[List["EulerGamma", " ", RowBox[List["(", RowBox[List["z", "+", "\[Nu]"]], ")"]]]], "-", RowBox[List["\[Nu]", " ", RowBox[List["PolyGamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]]]], ")"]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]]], "+", RowBox[List[SuperscriptBox["z", RowBox[List["1", "-", "\[Nu]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List["Hypergeometric2F1Regularized", "[", RowBox[List["1", ",", "2", ",", RowBox[List["2", "-", "\[Nu]"]], ",", RowBox[List["-", FractionBox["z", "k"]]]]], "]"]], SuperscriptBox["k", "2"]]]]]]]], "True"]], Rule[ColumnAlignments, List[Left]], Rule[ColumnSpacings, 1.2`], Rule[ColumnWidths, Automatic]]]]]]]]

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

2007-05-02

PolyGamma[z]