FourierCosTransform[Infinity, t, z] == DiracDelta[z] 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> <mrow> <msub> <mi> &#8497;&#119992; </mi> <mi> t </mi> </msub> <mo> [ </mo> <semantics> <mi> &#8734; </mi> <annotation-xml encoding='MathML-Content'> <infinity /> </annotation-xml> </semantics> <mo> ] </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mi> &#8734; </mi> <annotation-xml encoding='MathML-Content'> <infinity /> </annotation-xml> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> &#8497;&#119992; </ci> <ci> t </ci> </apply> <apply> <ci> DirectedInfinity </ci> <cn type='integer'> 1 </cn> </apply> </apply> <ci> z </ci> </apply> <apply> <times /> <apply> <ci> DiracDelta </ci> <ci> z </ci> </apply> <apply> <ci> DirectedInfinity </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["FourierCosTransform", "[", RowBox[List["\[Infinity]", ",", "t_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["DiracDelta", "[", "z", "]"]], " ", "\[Infinity]"]]]]]]

2001-10-29