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Min




Mathematica Notation

Traditional Notation









Elementary Functions > Min[x1 ,x2,...] > Integral transforms > Fourier exp transforms





http://functions.wolfram.com/01.35.22.0001.01









  


  








Input Form





FourierTransform[Min[Subscript[t, 1], Subscript[t, 2]], {Subscript[t, 1], Subscript[t, 2]}, {Subscript[z, 1], Subscript[z, 2]}] == DiracDelta[Subscript[z, 1] + Subscript[z, 2]]/Subscript[z, 1]^2 - I Pi (DiracDelta[Subscript[z, 2]] Derivative[1][DiracDelta][ Subscript[z, 1]] + DiracDelta[Subscript[z, 1]] Derivative[1][DiracDelta][Subscript[z, 2]])








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["FourierTransform", "[", RowBox[List[RowBox[List["Min", "[", RowBox[List[SubscriptBox["t", "1"], ",", SubscriptBox["t", "2"]]], "]"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["t", "1"], ",", SubscriptBox["t", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["z", "1"], ",", SubscriptBox["z", "2"]]], "}"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["DiracDelta", "[", RowBox[List[SubscriptBox["z", "1"], "+", SubscriptBox["z", "2"]]], "]"]], SubsuperscriptBox["z", "1", "2"]], "-", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["DiracDelta", "[", SubscriptBox["z", "2"], "]"]], " ", RowBox[List[SuperscriptBox["DiracDelta", "\[Prime]", Rule[MultilineFunction, None]], "[", SubscriptBox["z", "1"], "]"]]]], "+", RowBox[List[RowBox[List["DiracDelta", "[", SubscriptBox["z", "1"], "]"]], " ", RowBox[List[SuperscriptBox["DiracDelta", "\[Prime]", Rule[MultilineFunction, None]], "[", SubscriptBox["z", "2"], "]"]]]]]], ")"]]]]]]]]]]








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> <msub> <mi> &#8497; </mi> <mrow> <mo> { </mo> <mrow> <msub> <mi> t </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> t </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </msub> <mo> [ </mo> <mrow> <mi> min </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> t </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> t </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ] </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <msubsup> <mi> z </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> </mfrac> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> &#948; </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> &#948; </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <msub> <mi> &#8497; </mi> <mrow> <mo> { </mo> <mrow> <msub> <mi> t </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> t </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </msub> <mo> [ </mo> <mrow> <mi> min </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> t </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> t </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> ] </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <msubsup> <mi> z </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> </mfrac> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> &#948; </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> &#948; </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["FourierTransform", "[", RowBox[List[RowBox[List["Min", "[", RowBox[List[SubscriptBox["t_", "1"], ",", SubscriptBox["t_", "2"]]], "]"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["t_", "1"], ",", SubscriptBox["t_", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[SubscriptBox["z_", "1"], ",", SubscriptBox["z_", "2"]]], "}"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["DiracDelta", "[", RowBox[List[SubscriptBox["zz", "1"], "+", SubscriptBox["zz", "2"]]], "]"]], SubsuperscriptBox["zz", "1", "2"]], "-", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["DiracDelta", "[", SubscriptBox["zz", "2"], "]"]], " ", RowBox[List[SuperscriptBox["DiracDelta", "\[Prime]", Rule[MultilineFunction, None]], "[", SubscriptBox["zz", "1"], "]"]]]], "+", RowBox[List[RowBox[List["DiracDelta", "[", SubscriptBox["zz", "1"], "]"]], " ", RowBox[List[SuperscriptBox["DiracDelta", "\[Prime]", Rule[MultilineFunction, None]], "[", SubscriptBox["zz", "2"], "]"]]]]]], ")"]]]]]]]]]]


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





2001-10-29