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http://functions.wolfram.com/01.04.21.0045.01
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Integrate[(Log[t]^m Log[1 - t]^n)/t, {t, 0, 1}] ==
Limit[
Limit[
D[D[(1/\[Mu]) Exp[-Sum[Sum[(((-1)^(p + q) \[Lambda]^p \[Mu]^q
(p + q - 1)!)/(p! q!)) Zeta[p + q], {q, 1, m + 1}],
{p, 1, n + 1}]], {\[Mu], m}], {\[Lambda], n}], \[Mu] -> 0],
\[Lambda] -> 0] /; Element[m, Integers] && m >= 0 &&
Element[n, Integers] && n > 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "1"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Log", "[", "t", "]"]], "m"], " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["1", "-", "t"]], "]"]], "n"]]], "t"], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List["Limit", "[", RowBox[List[RowBox[List["Limit", "[", RowBox[List[RowBox[List["D", "[", RowBox[List[RowBox[List["D", "[", RowBox[List[RowBox[List[FractionBox["1", "\[Mu]"], RowBox[List["Exp", "[", RowBox[List["-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["p", "=", "1"]], RowBox[List["n", "+", "1"]]], RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "1"]], RowBox[List["m", "+", "1"]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["p", "+", "q"]]], SuperscriptBox["\[Lambda]", "p"], SuperscriptBox["\[Mu]", "q"], RowBox[List[RowBox[List["(", RowBox[List["p", "+", "q", "-", "1"]], ")"]], "!"]], " "]], RowBox[List[RowBox[List["p", "!"]], RowBox[List["q", "!"]]]]], RowBox[List["Zeta", "[", RowBox[List["p", "+", "q"]], "]"]]]]]], ")"]]]]]], "]"]]]], ",", RowBox[List["{", RowBox[List["\[Mu]", ",", "m"]], "}"]]]], "]"]], ",", RowBox[List["{", RowBox[List["\[Lambda]", ",", "n"]], "}"]]]], "]"]], ",", RowBox[List["\[Mu]", "\[Rule]", "0"]]]], "]"]], ",", RowBox[List["\[Lambda]", "\[Rule]", "0"]]]], "]"]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]], "\[And]", 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> <mn> 1 </mn> </msubsup> <mrow> <mfrac> <mrow> <mrow> <msup> <mi> log </mi> <mi> m </mi> </msup> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> log </mi> <mi> n </mi> </msup> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> t </mi> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> λ </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mn> 0 </mn> </mrow> </munder> <mo> ⁢ </mo> <mtext>   </mtext> <mrow> <munder> <mi> lim </mi> <mrow> <mi> μ </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mn> 0 </mn> </mrow> </munder> <mo> ⁢ </mo> <mtext>   </mtext> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mi> n </mi> </msup> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mi> m </mi> </msup> <mfrac> <mrow> <mi> exp </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> p </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> q </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> p </mi> <mo> + </mo> <mi> q </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mi> p </mi> </msup> <mo> ⁢ </mo> <msup> <mi> μ </mi> <mi> q </mi> </msup> <mo> ⁢ </mo> <mtext> </mtext> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> p </mi> <mo> + </mo> <mi> q </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> p </mi> <mo> + </mo> <mi> q </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["p", "+", "q"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> <mrow> <mrow> <mi> p </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> q </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> μ </mi> </mfrac> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> μ </mi> <mi> m </mi> </msup> </mrow> </mfrac> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> λ </mi> <mi> n </mi> </msup> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> m </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> <mo> ∧ </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </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='integer'> 1 </cn> </uplimit> <apply> <times /> <apply> <power /> <apply> <ln /> <ci> t </ci> </apply> <ci> m </ci> </apply> <apply> <power /> <apply> <ln /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> t </ci> </apply> </apply> </apply> <ci> n </ci> </apply> <apply> <power /> <ci> t </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <limit /> <bvar> <ci> λ </ci> </bvar> <condition> <apply> <tendsto /> <ci> λ </ci> <cn type='integer'> 0 </cn> </apply> </condition> <apply> <limit /> <bvar> <ci> μ </ci> </bvar> <condition> <apply> <tendsto /> <ci> μ </ci> <cn type='integer'> 0 </cn> </apply> </condition> <apply> <partialdiff /> <bvar> <ci> λ </ci> <degree> <ci> n </ci> </degree> </bvar> <apply> <partialdiff /> <bvar> <ci> μ </ci> <degree> <ci> m </ci> </degree> </bvar> <apply> <times /> <apply> <exp /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sum /> <bvar> <ci> q </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </uplimit> <apply> <sum /> <bvar> <ci> p </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> p </ci> <ci> q </ci> </apply> </apply> <apply> <power /> <ci> λ </ci> <ci> p </ci> </apply> <apply> <power /> <ci> μ </ci> <ci> q </ci> </apply> <apply> <factorial /> <apply> <plus /> <ci> p </ci> <ci> q </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Zeta </ci> <apply> <plus /> <ci> p </ci> <ci> q </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> p </ci> </apply> <apply> <factorial /> <ci> q </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <ci> μ </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> m </ci> <ci> ℕ </ci> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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