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 PolyGamma

 http://functions.wolfram.com/06.15.21.0002.01

 Input Form

 Integrate[z^(\[Alpha] - 1) PolyGamma[\[Nu], z], z] == (z^(\[Alpha] - \[Nu] - 1)/((\[Alpha] - \[Nu] - 1) Gamma[-\[Nu]])) (1/(\[Alpha] - \[Nu] - 1) + EulerGamma + PolyGamma[-\[Nu]] - Log[z]) + (z^(1 + \[Alpha] - \[Nu])/((1 + \[Alpha] - \[Nu]) Gamma[2 - \[Nu]])) Sum[(1/(k + 1)^2) HypergeometricPFQ[{1, 2, 1 + \[Alpha] - \[Nu]}, {2 - \[Nu], 2 + \[Alpha] - \[Nu]}, -(z/(k + 1))], {k, 0, Infinity}] - (EulerGamma z^(\[Alpha] - \[Nu]))/(Gamma[1 - \[Nu]] (\[Alpha] - \[Nu]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], RowBox[List["PolyGamma", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "\[Nu]", "-", "1"]]], " "]], RowBox[List[RowBox[List["(", RowBox[List["\[Alpha]", "-", "\[Nu]", "-", "1"]], ")"]], RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]]], RowBox[List["(", RowBox[List[FractionBox["1", RowBox[List["\[Alpha]", "-", "\[Nu]", "-", "1"]]], "+", " ", "EulerGamma", "+", RowBox[List["PolyGamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], "-", RowBox[List["Log", "[", "z", "]"]]]], ")"]]]], "+", RowBox[List[FractionBox[SuperscriptBox["z", RowBox[List["1", "+", "\[Alpha]", "-", "\[Nu]"]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[Alpha]", "-", "\[Nu]"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "-", "\[Nu]"]], "]"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox["1", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "2"], " "]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "2", ",", RowBox[List["1", "+", "\[Alpha]", "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "-", "\[Nu]"]], ",", RowBox[List["2", "+", "\[Alpha]", "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", FractionBox["z", RowBox[List["k", "+", "1"]]]]]]], "]"]]]]]]]], "-", FractionBox[RowBox[List["EulerGamma", " ", SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "\[Nu]"]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]], RowBox[List["(", RowBox[List["\[Alpha]", "-", "\[Nu]"]], ")"]]]]]]]]]]]

 MathML Form

 z α - 1 ψ TagBox["\[Psi]", PolyGamma] ( ν ) ( z ) z - TagBox["\[DoubledGamma]", Function[EulerGamma]] z α - ν Γ ( 1 - ν ) ( α - ν ) + z α - ν - 1 ( α - ν - 1 ) Γ ( - ν ) ( ψ TagBox["\[Psi]", PolyGamma] ( - ν ) + 1 α - ν - 1 + - log ( z ) TagBox[RowBox[List["\[DoubledGamma]", "-", RowBox[List["log", "(", "z", ")"]]]], Function[EulerGamma]] ) + z α - ν + 1 ( α - ν + 1 ) Γ ( 2 - ν ) k = 0 1 ( k + 1 ) 2 3 F 2 ( 1 , 2 , α - ν + 1 ; 2 - ν , α - ν + 2 ; - z k + 1 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["3", TraditionalForm]], SubscriptBox["F", FormBox["2", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox["2", HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Alpha]", "-", "\[Nu]", "+", "1"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["2", "-", "\[Nu]"]], HypergeometricPFQ, Rule[Editable, True]], ",", TagBox[RowBox[List["\[Alpha]", "-", "\[Nu]", "+", "2"]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], ";", TagBox[RowBox[List["-", FractionBox["z", RowBox[List["k", "+", "1"]]]]], HypergeometricPFQ, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] z z α -1 PolyGamma ν z -1 z α -1 ν Gamma 1 -1 ν α -1 ν -1 z α -1 ν -1 α -1 ν -1 Gamma -1 ν -1 PolyGamma -1 ν 1 α -1 ν -1 -1 z α -1 ν 1 α -1 ν 1 Gamma 2 -1 ν -1 k 0 1 k 1 2 -1 HypergeometricPFQ 1 2 α -1 ν 1 2 -1 ν α -1 ν 2 -1 z k 1 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", RowBox[List["PolyGamma", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "\[Nu]", "-", "1"]]], " ", RowBox[List["(", RowBox[List[FractionBox["1", RowBox[List["\[Alpha]", "-", "\[Nu]", "-", "1"]]], "+", "EulerGamma", "+", RowBox[List["PolyGamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], "-", RowBox[List["Log", "[", "z", "]"]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["\[Alpha]", "-", "\[Nu]", "-", "1"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List["1", "+", "\[Alpha]", "-", "\[Nu]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "2", ",", RowBox[List["1", "+", "\[Alpha]", "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "-", "\[Nu]"]], ",", RowBox[List["2", "+", "\[Alpha]", "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", FractionBox["z", RowBox[List["k", "+", "1"]]]]]]], "]"]], SuperscriptBox[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], "2"]]]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[Alpha]", "-", "\[Nu]"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["2", "-", "\[Nu]"]], "]"]]]]], "-", FractionBox[RowBox[List["EulerGamma", " ", SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "\[Nu]"]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Nu]"]], "]"]], " ", RowBox[List["(", RowBox[List["\[Alpha]", "-", "\[Nu]"]], ")"]]]]]]]]]]]

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

 2001-10-29