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http://functions.wolfram.com/14.04.21.0002.01
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Integrate[DiracDelta[Subscript[t, 1] - Subscript[a, 1],
Subscript[t, 2] - Subscript[a, 2], â¦,
Subscript[t, n] - Subscript[a, n]]*
f[Subscript[t, 1], Subscript[t, 2], â¦, Subscript[t, n]],
{Subscript[t, 1], -Subscript[d, 1], Subscript[d, 1]},
{Subscript[t, 2], -Subscript[d, 2], Subscript[d, 2]},
â¦,
{Subscript[t, n], -Subscript[d, n], Subscript[d, n]}] == 1/;
Inequality[-Infinity, LessEqual, -Subscript[d, k], Less, Subscript[a, k], Less,
Subscript[d, k], LessEqual, Infinity] && 1 <= k <= n
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", InterpretationBox[RowBox[List["-", SubscriptBox["d", "1"]]], DirectedInfinity[-1]], InterpretationBox[SubscriptBox["d", "1"], DirectedInfinity[1]]], RowBox[List[SubsuperscriptBox["\[Integral]", InterpretationBox[RowBox[List["-", SubscriptBox["d", "2"]]], DirectedInfinity[-1]], SubscriptBox["d", "2"]], RowBox[List["\[Ellipsis]", RowBox[List[SubsuperscriptBox["\[Integral]", InterpretationBox[RowBox[List["-", SubscriptBox["d", "n"]]], DirectedInfinity[-1]], InterpretationBox[SubscriptBox["d", "n"], DirectedInfinity[1]]], RowBox[List[RowBox[List["DiracDelta", "[", RowBox[List[RowBox[List[SubscriptBox["t", "1"], "-", SubscriptBox["a", "1"]]], ",", RowBox[List[SubscriptBox["t", "2"], "-", SubscriptBox["a", "2"]]], ",", "\[Ellipsis]", ",", RowBox[List[SubscriptBox["t", "n"], "-", SubscriptBox["a", "n"]]]]], "]"]], RowBox[List["f", "[", RowBox[List[SubscriptBox["t", "1"], ",", SubscriptBox["t", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["t", "n"]]], "]"]], RowBox[List["\[DifferentialD]", SubscriptBox["t", "1"]]], RowBox[List["\[DifferentialD]", SubscriptBox["t", "2"]]], " ", "\[Ellipsis]", RowBox[List["\[DifferentialD]", SubscriptBox["t", "n"]]]]]]]]]]]]], "\[Equal]", RowBox[List["f", "[", RowBox[List[SubscriptBox["a", "1"], ",", SubscriptBox["a", "2"], ",", "\[Ellipsis]", ",", SubscriptBox["a", "n"]]], "]"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["-", "\[Infinity]"]], "\[LessEqual]", RowBox[List["-", SubscriptBox["d", "k"]]], "<", SubscriptBox["a", "k"], "<", SubscriptBox["d", "k"], "\[LessEqual]", "\[Infinity]"]], "\[And]", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", "n"]]]]]]]]
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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> <msubsup> <mo> ∫ </mo> <semantics> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <times /> <cn type='integer'> -1 </cn> <infinity /> </apply> </annotation-xml> </semantics> <semantics> <mi> ∞ </mi> <annotation-xml encoding='MathML-Content'> <infinity /> </annotation-xml> </semantics> </msubsup> <mrow> <msubsup> <mo> ∫ </mo> <semantics> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <times /> <cn type='integer'> -1 </cn> <infinity /> </apply> </annotation-xml> </semantics> <msub> <mi> d </mi> <mn> 2 </mn> </msub> </msubsup> <mrow> <mo> … </mo> <mo> ⁢ </mo> <mrow> <msubsup> <mo> ∫ </mo> <semantics> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <times /> <cn type='integer'> -1 </cn> <infinity /> </apply> </annotation-xml> </semantics> <semantics> <mi> ∞ </mi> <annotation-xml encoding='MathML-Content'> <infinity /> </annotation-xml> </semantics> </msubsup> <mrow> <mrow> <mrow> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <msub> <mi> t </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mrow> <msub> <mi> t </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <msub> <mi> t </mi> <mi> n </mi> </msub> <mo> - </mo> <msub> <mi> a </mi> <mi> n </mi> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> f </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> t </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> t </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> t </mi> <mi> n </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <msub> <mi> t </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <msub> <mi> t </mi> <mn> 2 </mn> </msub> </mrow> <mo> ⁢ </mo> <mo> … </mo> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <msub> <mi> t </mi> <mi> n </mi> </msub> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mi> f </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> a </mi> <mi> n </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> <mo> ≤ </mo> <mrow> <mo> - </mo> <msub> <mi> d </mi> <mi> k </mi> </msub> </mrow> <mo> < </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> < </mo> <msub> <mi> d </mi> <mi> k </mi> </msub> <mo> ≤ </mo> <mi> ∞ </mi> </mrow> <mo> ∧ </mo> <mrow> <mn> 1 </mn> <mo> ≤ </mo> <mi> k </mi> <mo> ≤ </mo> <mi> n </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <msubsup> <mo> ∫ </mo> <semantics> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <times /> <cn type='integer'> -1 </cn> <infinity /> </apply> </annotation-xml> </semantics> <semantics> <mi> ∞ </mi> <annotation-xml encoding='MathML-Content'> <infinity /> </annotation-xml> </semantics> </msubsup> <mrow> <msubsup> <mo> ∫ </mo> <semantics> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <times /> <cn type='integer'> -1 </cn> <infinity /> </apply> </annotation-xml> </semantics> <msub> <mi> d </mi> <mn> 2 </mn> </msub> </msubsup> <mrow> <mo> … </mo> <mo> ⁢ </mo> <mrow> <msubsup> <mo> ∫ </mo> <semantics> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <times /> <cn type='integer'> -1 </cn> <infinity /> </apply> </annotation-xml> </semantics> <semantics> <mi> ∞ </mi> <annotation-xml encoding='MathML-Content'> <infinity /> </annotation-xml> </semantics> </msubsup> <mrow> <mrow> <mrow> <semantics> <mi> δ </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mrow> <msub> <mi> t </mi> <mn> 1 </mn> </msub> <mo> - </mo> <msub> <mi> a </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mrow> <msub> <mi> t </mi> <mn> 2 </mn> </msub> <mo> - </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> </mrow> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mrow> <msub> <mi> t </mi> <mi> n </mi> </msub> <mo> - </mo> <msub> <mi> a </mi> <mi> n </mi> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> f </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> t </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> t </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> t </mi> <mi> n </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <msub> <mi> t </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <msub> <mi> t </mi> <mn> 2 </mn> </msub> </mrow> <mo> ⁢ </mo> <mo> … </mo> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <msub> <mi> t </mi> <mi> n </mi> </msub> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mi> f </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msub> <mi> a </mi> <mn> 1 </mn> </msub> <mo> , </mo> <msub> <mi> a </mi> <mn> 2 </mn> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mi> a </mi> <mi> n </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> <mo> ≤ </mo> <mrow> <mo> - </mo> <msub> <mi> d </mi> <mi> k </mi> </msub> </mrow> <mo> < </mo> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo> < </mo> <msub> <mi> d </mi> <mi> k </mi> </msub> <mo> ≤ </mo> <mi> ∞ </mi> </mrow> <mo> ∧ </mo> <mrow> <mn> 1 </mn> <mo> ≤ </mo> <mi> k </mi> <mo> ≤ </mo> <mi> n </mi> </mrow> </mrow> </mrow> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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