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http://functions.wolfram.com/01.04.21.0073.01
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Integrate[t^(2 m) Log[2 Cos[t/2]]^n, {t, 0, Pi}] ==
Limit[Limit[Pi (-1)^m
D[D[E^(-Sum[\[Mu]^q Sum[(((-1)^r (1 + (-1)^q) (r + q - 1)! \[Lambda]^r)
Zeta[r + q])/(2^r r! q!), {r, 1, m + 1}], {q, 1, n + 1}]),
{\[Mu], 2 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", "\[Pi]"], RowBox[List[RowBox[List[SuperscriptBox["t", RowBox[List["2", " ", "m"]]], " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["2", " ", RowBox[List["Cos", "[", FractionBox["t", "2"], "]"]]]], "]"]], "n"]]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List["Limit", "[", RowBox[List[RowBox[List["Limit", "[", RowBox[List[RowBox[List["\[Pi]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["\[Lambda]", ",", "n"]], "}"]]], RowBox[List["(", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["\[Mu]", ",", RowBox[List["2", " ", "m"]]]], "}"]]], SuperscriptBox["\[ExponentialE]", RowBox[List["-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "1"]], RowBox[List["n", "+", "1"]]], RowBox[List[SuperscriptBox["\[Mu]", "q"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "1"]], RowBox[List["m", "+", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "r"], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "q"]]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["r", "+", "q", "-", "1"]], ")"]], "!"]], " ", SuperscriptBox["\[Lambda]", "r"]]], ")"]], " ", RowBox[List["Zeta", "[", RowBox[List["r", "+", "q"]], "]"]]]], RowBox[List[SuperscriptBox["2", "r"], " ", RowBox[List["r", "!"]], " ", RowBox[List["q", "!"]]]]]]]]]]]]]]]], ")"]]]]]], ",", RowBox[List["\[Mu]", "\[Rule]", "0"]]]], "]"]], ",", RowBox[List["\[Lambda]", "\[Rule]", "0"]]]], "]"]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", 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> <mi> π </mi> </msubsup> <mrow> <mrow> <msup> <mi> t </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msup> <mi> log </mi> <mi> n </mi> </msup> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mi> t </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <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> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> <mo> ⁢ </mo> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mi> n </mi> </msup> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> </msup> <msup> <mi> ⅇ </mi> <mrow> <mo> - </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> q </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </munderover> <mrow> <msup> <mi> μ </mi> <mi> q </mi> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> r </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> <mi> r </mi> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> q </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> + </mo> <mi> r </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> λ </mi> <mi> r </mi> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> q </mi> <mo> + </mo> <mi> r </mi> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox[RowBox[List["q", "+", "r"]], Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> <mrow> <msup> <mn> 2 </mn> <mi> r </mi> </msup> <mo> ⁢ </mo> <mrow> <mi> r </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> q </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> </mrow> </mrow> </msup> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> μ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> λ </mi> <mi> n </mi> </msup> </mrow> </mfrac> </mrow> </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> <pi /> </uplimit> <apply> <times /> <apply> <power /> <ci> t </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <cos /> <apply> <times /> <ci> t </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <ci> n </ci> </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> <times /> <pi /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <partialdiff /> <bvar> <ci> λ </ci> <degree> <ci> n </ci> </degree> </bvar> <apply> <partialdiff /> <bvar> <ci> μ </ci> <degree> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> </degree> </bvar> <apply> <power /> <exponentiale /> <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> n </ci> <cn type='integer'> 1 </cn> </apply> </uplimit> <apply> <times /> <apply> <power /> <ci> μ </ci> <ci> q </ci> </apply> <apply> <sum /> <bvar> <ci> r </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> m </ci> <cn type='integer'> 1 </cn> </apply> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> r </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <cn type='integer'> -1 </cn> <ci> q </ci> </apply> </apply> <apply> <factorial /> <apply> <plus /> <ci> q </ci> <ci> r </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> λ </ci> <ci> r </ci> </apply> <apply> <ci> Zeta </ci> <apply> <plus /> <ci> q </ci> <ci> r </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> r </ci> </apply> <apply> <factorial /> <ci> r </ci> </apply> <apply> <factorial /> <ci> q </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </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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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "0", "\[Pi]"], RowBox[List[RowBox[List[SuperscriptBox["t_", RowBox[List["2", " ", "m_"]]], " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["2", " ", RowBox[List["Cos", "[", FractionBox["t_", "2"], "]"]]]], "]"]], "n_"]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Limit", "[", RowBox[List[RowBox[List["Limit", "[", RowBox[List[RowBox[List["\[Pi]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["\[Lambda]", ",", "n"]], "}"]]]]], RowBox[List["(", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["\[Mu]", ",", RowBox[List["2", " ", "m"]]]], "}"]]]]], SuperscriptBox["\[ExponentialE]", RowBox[List["-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "1"]], RowBox[List["n", "+", "1"]]], RowBox[List[SuperscriptBox["\[Mu]", "q"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["r", "=", "1"]], RowBox[List["m", "+", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "r"], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "q"]]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["r", "+", "q", "-", "1"]], ")"]], "!"]], " ", SuperscriptBox["\[Lambda]", "r"]]], ")"]], " ", RowBox[List["Zeta", "[", RowBox[List["r", "+", "q"]], "]"]]]], RowBox[List[SuperscriptBox["2", "r"], " ", RowBox[List["r", "!"]], " ", RowBox[List["q", "!"]]]]]]]]]]]]]]]], ")"]]]]]], ",", RowBox[List["\[Mu]", "\[Rule]", "0"]]]], "]"]], ",", RowBox[List["\[Lambda]", "\[Rule]", "0"]]]], "]"]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]], "&&", 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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){kind=small}
