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http://functions.wolfram.com/01.04.21.0044.01
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Integrate[Log[t/(1 - t)]^n/(1 - t), {t, 0, 1/2}] ==
(-1)^n n! Zeta[n + 1] (1 - 2^(-n)) /; Element[n, Integers] && n > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", FractionBox["1", "2"]], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["Log", "[", FractionBox["t", RowBox[List["1", "-", "t"]]], "]"]], "n"], RowBox[List["1", "-", "t"]]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["n", "!"]], " ", RowBox[List["Zeta", "[", RowBox[List["n", "+", "1"]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["2", RowBox[List["-", "n"]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </msubsup> <mrow> <mfrac> <mrow> <msup> <mi> log </mi> <mi> n </mi> </msup> <mo> ( </mo> <mfrac> <mi> t </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> t </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> t </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> ⁢ </mo> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["n", "+", "1"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mn> 2 </mn> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <cn type='rational'> 1 <sep /> 2 </cn> </uplimit> <apply> <times /> <apply> <power /> <apply> <ln /> <apply> <times /> <ci> t </ci> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> n </ci> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <factorial /> <ci> n </ci> </apply> <apply> <ci> Zeta </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", FractionBox["1", "2"]], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["Log", "[", FractionBox["t_", RowBox[List["1", "-", "t_"]]], "]"]], "n_"], RowBox[List["1", "-", "t_"]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["n", "!"]], " ", RowBox[List["Zeta", "[", RowBox[List["n", "+", "1"]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["2", RowBox[List["-", "n"]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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