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http://functions.wolfram.com/04.06.22.0002.01
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FourierCosTransform[Mod[t, n], t, z] == n Sqrt[Pi/2] DiracDelta[z] +
(n z Cot[(n z)/2] - 2)/(Sqrt[2 Pi] z^2)
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Cell[BoxData[RowBox[List[RowBox[List["FourierCosTransform", "[", RowBox[List[RowBox[List["Mod", "[", RowBox[List["t", ",", "n"]], "]"]], ",", "t", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["n", " ", SqrtBox[FractionBox["\[Pi]", "2"]], " ", RowBox[List["DiracDelta", "[", "z", "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["n", " ", "z", " ", RowBox[List["Cot", "[", FractionBox[RowBox[List["n", " ", "z"]], "2"], "]"]]]], "-", "2"]], RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", SuperscriptBox["z", "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> <mrow> <msub> <mi> ℱ𝒸 </mi> <mi> t </mi> </msub> <mo> [ </mo> <semantics> <mrow> <mi> t </mi> <mo> ⁢ </mo> <mi> mod </mi> <mo> ⁢ </mo> <mi> n </mi> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <ci> $CellContext`t </ci> <ci> $CellContext`n </ci> </apply> </annotation-xml> </semantics> <mo> ] </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mrow> <msqrt> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </msqrt> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <mrow> <mi> cot </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> n </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msqrt> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </msqrt> <mo> ⁢ </mo> <mi> n </mi> <mo> ⁢ </mo> <mrow> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> ℱ𝒸 </ci> <ci> t </ci> </apply> <apply> <rem /> <ci> $CellContext`t </ci> <ci> $CellContext`n </ci> </apply> </apply> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> n </ci> <ci> z </ci> <apply> <cot /> <apply> <times /> <ci> n </ci> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> n </ci> <apply> <ci> DiracDelta </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["FourierCosTransform", "[", RowBox[List[RowBox[List["Mod", "[", RowBox[List["t_", ",", "n_"]], "]"]], ",", "t_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["n", " ", SqrtBox[FractionBox["\[Pi]", "2"]], " ", RowBox[List["DiracDelta", "[", "z", "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["n", " ", "z", " ", RowBox[List["Cot", "[", FractionBox[RowBox[List["n", " ", "z"]], "2"], "]"]]]], "-", "2"]], RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", SuperscriptBox["z", "2"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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