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 Log

 http://functions.wolfram.com/01.04.21.0076.01

 Input Form

 Integrate[(z^(\[Alpha] - 1) Log[z]^n)/(1 - z), z] == z^\[Alpha] Sum[Binomial[n, j] (-1)^j j! Log[z]^(n - j) \[Alpha]^(-j - 1) HypergeometricPFQ[{1, Subscript[a, 1], \[Ellipsis], Subscript[a, j + 1]}, {1 + Subscript[a, 1], \[Ellipsis], 1 + Subscript[a, j + 1]}, z], {j, 0, n}] /; Subscript[a, 1] == Subscript[a, 2] == \[Ellipsis] == Subscript[a, n + 1] == \[Alpha] && Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List["\[Alpha]", "-", "1"]]], " ", SuperscriptBox[RowBox[List["Log", "[", "z", "]"]], "n"]]], RowBox[List["1", "-", "z"]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["j", "!"]], " ", SuperscriptBox[RowBox[List["Log", "[", "z", "]"]], RowBox[List["n", "-", "j"]]], " ", SuperscriptBox["\[Alpha]", RowBox[List[RowBox[List["-", "j"]], "-", "1"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", SubscriptBox["a", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["a", RowBox[List["j", "+", "1"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", SubscriptBox["a", "1"]]], ",", "\[Ellipsis]", ",", RowBox[List["1", "+", SubscriptBox["a", RowBox[List["j", "+", "1"]]]]]]], "}"]], ",", "z"]], "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", "1"], "\[Equal]", SubscriptBox["a", "2"], "\[Equal]", "\[Ellipsis]", "\[Equal]", SubscriptBox["a", RowBox[List["n", "+", "1"]]], "\[Equal]", "\[Alpha]"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 z α - 1 log n ( z ) 1 - z z z α j = 0 n ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( - 1 ) j j ! log n - j ( z ) α - j - 1 j + 2 F j + 1 ( 1 , α , , α , 1 ; α + 1 , , α + 1 ; z ) TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox[RowBox[List["j", "+", "2"]], TraditionalForm]], SubscriptBox["F", FormBox[RowBox[List["j", "+", "1"]], TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ], ",", "\[Alpha]", ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox["\[Alpha]", HypergeometricPFQ], ",", TagBox["1", HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox[TagBox[RowBox[List[TagBox[RowBox[List["\[Alpha]", "+", "1"]], HypergeometricPFQ], ",", TagBox["\[Ellipsis]", HypergeometricPFQ], ",", TagBox[RowBox[List["\[Alpha]", "+", "1"]], HypergeometricPFQ]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ], ";", TagBox["z", HypergeometricPFQ]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]]] /; n TagBox["\[DoubleStruckCapitalN]", Function[List[], Integers]] Condition z z α -1 z n 1 -1 z -1 z α j 0 n Binomial n j -1 j j z n -1 j α -1 j -1 HypergeometricPFQ 1 α 1 α 1 α 1 z n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z_", RowBox[List["\[Alpha]_", "-", "1"]]], " ", SuperscriptBox[RowBox[List["Log", "[", "z_", "]"]], "n_"]]], RowBox[List["1", "-", "z_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["z", "\[Alpha]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "j"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["j", "!"]], " ", SuperscriptBox[RowBox[List["Log", "[", "z", "]"]], RowBox[List["n", "-", "j"]]], " ", SuperscriptBox["\[Alpha]", RowBox[List[RowBox[List["-", "j"]], "-", "1"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["Join", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["Table", "[", RowBox[List["\[Alpha]", ",", RowBox[List["{", RowBox[List["K\$1", ",", "1", ",", RowBox[List["1", "+", "j"]]]], "}"]]]], "]"]]]], "]"]], ",", RowBox[List["Join", "[", RowBox[List["Table", "[", RowBox[List[RowBox[List["1", "+", "\[Alpha]"]], ",", RowBox[List["{", RowBox[List["K\$1", ",", "1", ",", RowBox[List["1", "+", "j"]]]], "}"]]]], "]"]], "]"]], ",", "z"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02