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http://functions.wolfram.com/05.13.21.0002.01
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Integrate[EulerE[m, t] EulerE[n, t], {t, 0, 1}] ==
(((-1)^n 4 (2^(m + n + 2) - 1) m! n!)/(m + n + 2)!) BernoulliB[m + n + 2]
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Cell[BoxData[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "0", "1"], RowBox[List[RowBox[List[RowBox[List["EulerE", "[", RowBox[List["m", ",", "t"]], "]"]], " ", RowBox[List["EulerE", "[", RowBox[List["n", ",", "t"]], "]"]]]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", "4", " ", RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["m", "+", "n", "+", "2"]]], "-", "1"]], ")"]], " ", RowBox[List["m", "!"]], " ", RowBox[List["n", "!"]], " "]], RowBox[List[RowBox[List["(", RowBox[List["m", "+", "n", "+", "2"]], ")"]], "!"]]], RowBox[List["BernoulliB", "[", RowBox[List["m", "+", "n", "+", "2"]], "]"]]]]]]]]
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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> <msubsup> <mo> ∫ </mo> <mn> 0 </mn> <mn> 1 </mn> </msubsup> <mrow> <mrow> <mrow> <msub> <semantics> <mi> E </mi> <annotation encoding='Mathematica'> TagBox["E", EulerE] </annotation> </semantics> <mi> m </mi> </msub> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <semantics> <mi> E </mi> <annotation encoding='Mathematica'> TagBox["E", EulerE] </annotation> </semantics> <mi> n </mi> </msub> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> ⁢ </mo> <mn> 4 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> m </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mtext> </mtext> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox["B", BernoulliB] </annotation> </semantics> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </msub> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <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> <ci> EulerE </ci> <ci> m </ci> <ci> t </ci> </apply> <apply> <ci> EulerE </ci> <ci> n </ci> <ci> t </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 4 </cn> <apply> <plus /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> m </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <factorial /> <ci> m </ci> </apply> <apply> <factorial /> <ci> n </ci> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> m </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> BernoulliB </ci> <apply> <plus /> <ci> m </ci> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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