FourierCosTransform[Gamma[a, t], t, x] == Gamma[a] Sqrt[Pi/2] DiracDelta[x] + (Gamma[a] Sqrt[2/Pi] (1/x) Sin[a ArcTan[x]])/(1 + x^2)^(a/2) /; Element[x, \[DoubleStruckCapitalR]] && Re[a] > -1

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["FourierCosTransform", "[", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["a", ",", "t"]], "]"]], ",", "t", ",", "x"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], SqrtBox[FractionBox["\[Pi]", "2"]], RowBox[List["DiracDelta", "[", "x", "]"]]]], "+", " ", RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", FractionBox["1", "x"], SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["x", "2"]]], ")"]], RowBox[List[RowBox[List["-", "a"]], "/", "2"]]], RowBox[List["Sin", "[", RowBox[List["a", " ", RowBox[List["ArcTan", "[", "x", "]"]]]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "\[DoubleStruckCapitalR]"]], "\[And]", RowBox[List[RowBox[List["Re", "[", "a", "]"]], ">", RowBox[List["-", "1"]]]]]]]]]]

<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;&#119992; </mi> <mi> t </mi> </msub> <mo> [ </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> , </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> &#10869; </mo> <mtext> </mtext> <mrow> <mrow> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mfrac> <mn> 2 </mn> <mi> &#960; </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mfrac> <mi> a </mi> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> x </mi> </mfrac> </mrow> <mo> + </mo> <mrow> <msqrt> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </msqrt> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> x </mi> <mo> &#8712; </mo> <mi> &#8477; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> &#8497;&#119992; </ci> <ci> t </ci> </apply> <apply> <ci> Gamma </ci> <ci> a </ci> <ci> t </ci> </apply> </apply> <ci> x </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> Gamma </ci> <ci> a </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> a </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <sin /> <apply> <times /> <ci> a </ci> <apply> <arctan /> <ci> x </ci> </apply> </apply> </apply> <apply> <power /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> </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> <apply> <ci> Gamma </ci> <ci> a </ci> </apply> <apply> <ci> DiracDelta </ci> <ci> x </ci> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> x </ci> <ci> &#8477; </ci> </apply> <apply> <gt /> <apply> <real /> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["FourierCosTransform", "[", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["a_", ",", "t_"]], "]"]], ",", "t_", ",", "x_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", SqrtBox[FractionBox["\[Pi]", "2"]], " ", RowBox[List["DiracDelta", "[", "x", "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["Gamma", "[", "a", "]"]], " ", SqrtBox[FractionBox["2", "\[Pi]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["x", "2"]]], ")"]], RowBox[List["-", FractionBox["a", "2"]]]], " ", RowBox[List["Sin", "[", RowBox[List["a", " ", RowBox[List["ArcTan", "[", "x", "]"]]]], "]"]]]], "x"]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "\[DoubleStruckCapitalR]"]], "&&", RowBox[List[RowBox[List["Re", "[", "a", "]"]], ">", RowBox[List["-", "1"]]]]]]]]]]]]

2001-10-29