DiracDelta[Subscript[x, 1], Subscript[x, 2], \[Ellipsis]] == (1/Pi) Limit[\[CurlyEpsilon]/(\[CurlyEpsilon]^2 + Subscript[x, 1]^2 + Subscript[x, 2]^2 + \[Ellipsis]), \[CurlyEpsilon] -> 0] /; Element[Subscript[x, k], Reals]

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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> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <msub> <mi> x </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> x </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mn> 1 </mn> <mi> &#960; </mi> </mfrac> <mo> &#8290; </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> &#949; </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mn> 0 </mn> </mrow> </munder> <mo> &#8290; </mo> <mtext> &#8201; </mtext> <mfrac> <mi> &#949; </mi> <mrow> <msup> <mi> &#949; </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msubsup> <mi> x </mi> <mn> 1 </mn> <mn> 2 </mn> </msubsup> <mo> + </mo> <msubsup> <mi> x </mi> <mn> 2 </mn> <mn> 2 </mn> </msubsup> <mo> + </mo> <mo> &#8230; </mo> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <msub> <mi> x </mi> <mi> k </mi> </msub> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> DiracDelta </ci> <apply> <ci> Subscript </ci> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <ci> &#8230; </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <limit /> <bvar> <ci> &#949; </ci> </bvar> <condition> <apply> <tendsto /> <ci> &#949; </ci> <cn type='integer'> 0 </cn> </apply> </condition> <apply> <times /> <ci> &#949; </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> &#949; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> x </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <ci> &#8230; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <apply> <ci> Subscript </ci> <ci> x </ci> <ci> k </ci> </apply> <reals /> </apply> </apply> </annotation-xml> </semantics> </math>

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2001-10-29