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variants of this functions
DiracDelta




Mathematica Notation

Traditional Notation









Generalized Functions > DiracDelta[x] > Series representations > Exponential Fourier series





http://functions.wolfram.com/14.03.06.0002.01









  


  








Input Form





DiracDelta[x] == 1/(2 Pi) + (1/Pi) Sum[Cos[k x], {k, 1, Infinity}] /; -Pi < x < Pi








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["DiracDelta", "[", "x", "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["2", " ", "\[Pi]"]]], "+", RowBox[List[FractionBox["1", "\[Pi]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List["Cos", "[", RowBox[List["k", " ", "x"]], "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List["-", "\[Pi]"]], "<", "x", "<", "\[Pi]"]]]]]]








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> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mi> &#960; </mi> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> - </mo> <mtext> </mtext> <mi> &#960; </mi> </mrow> <mo> &lt; </mo> <mi> x </mi> <mo> &lt; </mo> <mi> &#960; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> DiracDelta </ci> <ci> x </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <pi /> <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> <cos /> <apply> <times /> <ci> k </ci> <ci> x </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <lt /> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> <ci> x </ci> <pi /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["DiracDelta", "[", "x_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["2", " ", "\[Pi]"]]], "+", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], RowBox[List["Cos", "[", RowBox[List["k", " ", "x"]], "]"]]]], "\[Pi]"]]], "/;", RowBox[List[RowBox[List["-", "\[Pi]"]], "<", "x", "<", "\[Pi]"]]]]]]]]


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





2001-10-29