Sum[DiracDelta[x - t k], {k, -Infinity, Infinity}] == (1/t) Sum[Exp[(2 k Pi x I)/t], {k, -Infinity, Infinity}]

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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> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> x </mi> <mo> - </mo> <mrow> <mi> t </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#63449; </mo> <mrow> <mfrac> <mn> 1 </mn> <mi> t </mi> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> </mrow> <mi> &#8734; </mi> </munderover> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> x </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mi> t </mi> </mfrac> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <infinity /> </apply> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <ci> DiracDelta </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> t </ci> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> t </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <infinity /> </apply> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> <pi /> <ci> x </ci> <imaginaryi /> <apply> <power /> <ci> t </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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2007-05-02