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http://functions.wolfram.com/05.14.22.0002.01
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InverseFourierTransform[BernoulliB[n, t], t, x] ==
Sqrt[2 Pi] Sum[Binomial[n, k] BernoulliB[n - k] I^k
Derivative[k][DiracDelta][x], {k, 0, n}]
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Cell[BoxData[RowBox[List[RowBox[List["InverseFourierTransform", "[", RowBox[List[RowBox[List["BernoulliB", "[", RowBox[List["n", ",", "t"]], "]"]], ",", "t", ",", "x"]], "]"]], "\[Equal]", RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], RowBox[List["BernoulliB", "[", RowBox[List["n", "-", "k"]], "]"]], " ", SuperscriptBox["\[ImaginaryI]", "k"], " ", RowBox[List[SuperscriptBox["DiracDelta", TagBox[RowBox[List["(", "k", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "x", "]"]]]]]]]]]]]]
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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> <mi> ℱ </mi> <mi> t </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msubsup> <mo> [ </mo> <mrow> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox["B", BernoulliB] </annotation> </semantics> <mi> n </mi> </msub> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> ] </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <msqrt> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> n </mi> </mtd> </mtr> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> ⁢ </mo> <msub> <semantics> <mi> B </mi> <annotation encoding='Mathematica'> TagBox["B", BernoulliB] </annotation> </semantics> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </msub> <mo> ⁢ </mo> <msup> <mi> ⅈ </mi> <mi> k </mi> </msup> <mo> ⁢ </mo> <mrow> <msup> <mi> δ </mi> <semantics> <mrow> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", "k", ")"]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> InverseFourierTransform </ci> <apply> <ci> BernoulliB </ci> <ci> n </ci> <ci> t </ci> </apply> <ci> t </ci> <ci> x </ci> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <ci> Binomial </ci> <ci> n </ci> <ci> k </ci> </apply> <apply> <ci> BernoulliB </ci> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <apply> <power /> <imaginaryi /> <ci> k </ci> </apply> <apply> <partialdiff /> <bvar> <ci> x </ci> <degree> <ci> k </ci> </degree> </bvar> <apply> <ci> δ </ci> <ci> x </ci> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["InverseFourierTransform", "[", RowBox[List[RowBox[List["BernoulliB", "[", RowBox[List["n_", ",", "t_"]], "]"]], ",", "t_", ",", "x_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["BernoulliB", "[", RowBox[List["n", "-", "k"]], "]"]], " ", SuperscriptBox["\[ImaginaryI]", "k"], " ", RowBox[List[SuperscriptBox["DiracDelta", TagBox[RowBox[List["(", "k", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "x", "]"]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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