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Sin






Mathematica Notation

Traditional Notation









Elementary Functions > Sin[z] > Integral transforms > Fourier exp transforms





http://functions.wolfram.com/01.06.22.0001.01









  


  










Input Form





FourierTransform[Sin[a t], t, x] == I Sqrt[Pi/2] DiracDelta[-a + x] - I Sqrt[Pi/2] DiracDelta[a + x] /; Element[a, Reals]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["FourierTransform", "[", RowBox[List[RowBox[List["Sin", "[", RowBox[List["a", " ", "t"]], "]"]], ",", "t", ",", "x"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SqrtBox[FractionBox["\[Pi]", "2"]], " ", RowBox[List["DiracDelta", "[", RowBox[List[RowBox[List["-", "a"]], "+", "x"]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox[FractionBox["\[Pi]", "2"]], " ", RowBox[List["DiracDelta", "[", RowBox[List["a", "+", "x"]], "]"]]]]]]]], "/;", RowBox[List["a", "\[Element]", "Reals"]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mrow> <msub> <mi> &#8497; </mi> <mi> t </mi> </msub> <mo> [ </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> t </mi> </mrow> <mo> ) </mo> </mrow> <mo> ] </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#63449; </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </msqrt> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> x </mi> <mo> - </mo> <mi> a </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </msqrt> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> + </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> a </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[List[], Reals]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> FourierTransform </ci> <apply> <sin /> <apply> <times /> <ci> a </ci> <ci> t </ci> </apply> </apply> <ci> t </ci> <ci> x </ci> </apply> <apply> <plus /> <apply> <times /> <imaginaryi /> <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> <ci> DiracDelta </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <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> <ci> DiracDelta </ci> <apply> <plus /> <ci> a </ci> <ci> x </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> a </ci> <reals /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["FourierTransform", "[", RowBox[List[RowBox[List["Sin", "[", RowBox[List["a_", " ", "t_"]], "]"]], ",", "t_", ",", "x_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["\[ImaginaryI]", " ", SqrtBox[FractionBox["\[Pi]", "2"]], " ", RowBox[List["DiracDelta", "[", RowBox[List[RowBox[List["-", "a"]], "+", "x"]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox[FractionBox["\[Pi]", "2"]], " ", RowBox[List["DiracDelta", "[", RowBox[List["a", "+", "x"]], "]"]]]]]], "/;", RowBox[List["a", "\[Element]", "Reals"]]]]]]]]










Date Added to functions.wolfram.com (modification date)





2001-10-29