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http://functions.wolfram.com/12.01.22.0004.01
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FourierSinTransform[Abs[t], t, z] ==
(-Sqrt[Pi/2]) Derivative[1][DiracDelta][z]
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Cell[BoxData[RowBox[List[RowBox[List["FourierSinTransform", "[", RowBox[List[RowBox[List["Abs", "[", "t", "]"]], ",", "t", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", SqrtBox[FractionBox["\[Pi]", "2"]]]], " ", RowBox[List[SuperscriptBox["DiracDelta", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]]]]]]
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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> ℱ𝓈 </mi> <mi> t </mi> </msub> <mo> [ </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> t </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> ] </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <msqrt> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </msqrt> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> δ </mi> <mo> ′ </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> ℱ𝓈 </ci> <ci> t </ci> </apply> <apply> <abs /> <ci> t </ci> </apply> </apply> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <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> </apply> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> δ </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["FourierSinTransform", "[", RowBox[List[RowBox[List["Abs", "[", "t_", "]"]], ",", "t_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", SqrtBox[FractionBox["\[Pi]", "2"]]]], " ", RowBox[List[SuperscriptBox["DiracDelta", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
