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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<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> &#8747; </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> &#8290; </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> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> &#955; </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mn> 0 </mn> </mrow> </munder> <mo> &#8290; </mo> <mtext> &#8201; </mtext> <mrow> <munder> <mi> lim </mi> <mrow> <mi> &#956; </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mn> 0 </mn> </mrow> </munder> <mo> &#8290; </mo> <mtext> &#8201; </mtext> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mi> n </mi> </msup> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mi> m </mi> </msup> <mfrac> <mrow> <mi> exp </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mrow> <munderover> <mo> &#8721; </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> &#8721; </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> &#8290; </mo> <msup> <mi> &#955; </mi> <mi> p </mi> </msup> <mo> &#8290; </mo> <msup> <mi> &#956; </mi> <mi> q </mi> </msup> <mo> &#8290; </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> ! 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2001-10-29