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http://functions.wolfram.com/04.05.22.0001.01
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FourierTransform[FractionalPart[t], t, z] ==
I/(Sqrt[2 Pi] z) - (I/Sqrt[2 Pi])
Sum[(DiracDelta[2 k Pi - z] - DiracDelta[2 k Pi + z])/k, {k, 1, Infinity}]
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Cell[BoxData[RowBox[List[RowBox[List["FourierTransform", "[", RowBox[List[RowBox[List["FractionalPart", "[", "t", "]"]], ",", "t", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["\[ImaginaryI]", RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", "z"]]], "-", RowBox[List[FractionBox["\[ImaginaryI]", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[" ", RowBox[List[RowBox[List["DiracDelta", "[", RowBox[List[RowBox[List["2", " ", "k", " ", "\[Pi]"]], "-", "z"]], "]"]], "-", RowBox[List["DiracDelta", "[", RowBox[List[RowBox[List["2", " ", "k", " ", "\[Pi]"]], "+", "z"]], "]"]]]]]], "k"]]]]]]]]]]]
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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> ℱ </mi> <mi> t </mi> </msub> <mo> [ </mo> <mrow> <mi> frac </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> ] </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mi> ⅈ </mi> <mrow> <msqrt> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </msqrt> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mfrac> <mo> - </mo> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mtext> </mtext> </mrow> <msqrt> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mrow> <mrow> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> k </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> + </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> k </mi> </mfrac> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> FourierTransform </ci> <apply> <ci> FractionalPart </ci> <ci> t </ci> </apply> <ci> t </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <plus /> <apply> <ci> DiracDelta </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> <pi /> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> DiracDelta </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <ci> k </ci> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["FourierTransform", "[", RowBox[List[RowBox[List["FractionalPart", "[", "t_", "]"]], ",", "t_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["\[ImaginaryI]", RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", "z"]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["DiracDelta", "[", RowBox[List[RowBox[List["2", " ", "k", " ", "\[Pi]"]], "-", "z"]], "]"]], "-", RowBox[List["DiracDelta", "[", RowBox[List[RowBox[List["2", " ", "k", " ", "\[Pi]"]], "+", "z"]], "]"]]]], "k"]]]]], SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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