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BesselY






Mathematica Notation

Traditional Notation









Bessel-Type Functions > BesselY[nu,z] > Integration > Definite integration > Involving the direct function





http://functions.wolfram.com/03.03.21.0129.01









  


  










Input Form





Integrate[Subscript[Z, \[Nu]][a t, a \[Beta]] Subscript[Z, \[Nu]][b t, b \[Beta]] t, {t, \[Beta], Infinity}] == (1/a) DiracDelta[a - b] /; Subscript[Z, \[Nu]][x, y] == (BesselJ[\[Nu], y] BesselY[\[Nu], x] - BesselJ[\[Nu], x] BesselY[\[Nu], y])/ Sqrt[BesselJ[\[Nu], y]^2 + BesselY[\[Nu], y]^2] && Element[\[Nu], Reals] && Element[\[Beta], Reals] && \[Beta] >= 0 && Element[a, Reals] && Element[b, Reals]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", "\[Beta]", "\[Infinity]"], RowBox[List[RowBox[List[SubscriptBox["Z", "\[Nu]"], "[", RowBox[List[RowBox[List["a", " ", "t"]], ",", RowBox[List["a", " ", "\[Beta]"]]]], "]"]], RowBox[List[SubscriptBox["Z", "\[Nu]"], "[", RowBox[List[RowBox[List["b", " ", "t"]], ",", RowBox[List["b", " ", "\[Beta]"]]]], "]"]], "t", RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "a"], RowBox[List["DiracDelta", "[", RowBox[List["a", "-", "b"]], "]"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["Z", "\[Nu]"], "[", RowBox[List["x", ",", "y"]], "]"]], "\[Equal]", FractionBox[RowBox[List[RowBox[List[RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", "y"]], "]"]], RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]]]], "-", RowBox[List[RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]], RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "y"]], "]"]]]]]], SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", "y"]], "]"]], "2"], "+", SuperscriptBox[RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "y"]], "]"]], "2"]]]]]]], "\[And]", RowBox[List["\[Nu]", "\[Element]", "Reals"]], "\[And]", RowBox[List["\[Beta]", "\[Element]", "Reals"]], "\[And]", RowBox[List["\[Beta]", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["a", "\[Element]", "Reals"]], "\[And]", RowBox[List["b", "\[Element]", "Reals"]]]]]]]]










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> <msubsup> <mo> &#8747; </mo> <mi> &#946; </mi> <mi> &#8734; </mi> </msubsup> <mrow> <mrow> <mrow> <msub> <mi> Z </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> t </mi> </mrow> <mo> , </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> &#946; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> Z </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> t </mi> </mrow> <mo> , </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> &#946; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> t </mi> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> a </mi> <mo> - </mo> <mi> b </mi> </mrow> <mo> ) </mo> </mrow> <mi> a </mi> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <msub> <mi> Z </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mrow> <mi> x </mi> <mo> , </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <mrow> <mrow> <msub> <mi> J </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> Y </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <msub> <mi> J </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> Y </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <msqrt> <mrow> <msup> <mrow> <msub> <mi> J </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mrow> <msub> <mi> Y </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> </mrow> <mo> &#8743; 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</ci> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> Z </ci> <ci> &#957; </ci> </apply> <apply> <times /> <ci> a </ci> <ci> t </ci> </apply> <apply> <times /> <ci> a </ci> <ci> &#946; </ci> </apply> </apply> <apply> <apply> <ci> Subscript </ci> <ci> Z </ci> <ci> &#957; </ci> </apply> <apply> <times /> <ci> b </ci> <ci> t </ci> </apply> <apply> <times /> <ci> b </ci> <ci> &#946; </ci> </apply> </apply> <ci> t </ci> </apply> </apply> <apply> <times /> <apply> <ci> DiracDelta </ci> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> Z </ci> <ci> &#957; </ci> </apply> <ci> x </ci> <ci> y </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <ci> BesselJ </ci> <ci> &#957; </ci> <ci> y </ci> </apply> <apply> <ci> BesselY </ci> <ci> &#957; </ci> <ci> x </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> BesselJ </ci> <ci> &#957; </ci> <ci> x </ci> </apply> <apply> <ci> BesselY </ci> <ci> &#957; </ci> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <ci> BesselJ </ci> <ci> &#957; </ci> <ci> y </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> BesselY </ci> <ci> &#957; </ci> <ci> y </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <in /> <ci> &#957; </ci> <reals /> </apply> <apply> <in /> <ci> &#946; </ci> <reals /> </apply> <apply> <geq /> <ci> &#946; </ci> <cn type='integer'> 0 </cn> </apply> <apply> <in /> <ci> a </ci> <reals /> </apply> <apply> <in /> <ci> b </ci> <reals /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", "\[Beta]_", "\[Infinity]"], RowBox[List[RowBox[List[RowBox[List[SubscriptBox["Z", "\[Nu]"], "[", RowBox[List[RowBox[List["a_", " ", "t_"]], ",", RowBox[List["a_", " ", "\[Beta]_"]]]], "]"]], " ", RowBox[List[SubscriptBox["Z", "\[Nu]"], "[", RowBox[List[RowBox[List["b_", " ", "t_"]], ",", RowBox[List["b_", " ", "\[Beta]_"]]]], "]"]], " ", "t_"]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["DiracDelta", "[", RowBox[List["a", "-", "b"]], "]"]], "a"], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["Z", "\[Nu]"], "[", RowBox[List["x", ",", "y"]], "]"]], "\[Equal]", FractionBox[RowBox[List[RowBox[List[RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", "y"]], "]"]], " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]]]], "-", RowBox[List[RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]], " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "y"]], "]"]]]]]], SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", "y"]], "]"]], "2"], "+", SuperscriptBox[RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "y"]], "]"]], "2"]]]]]]], "&&", RowBox[List["\[Nu]", "\[Element]", "Reals"]], "&&", RowBox[List["\[Beta]", "\[Element]", "Reals"]], "&&", RowBox[List["\[Beta]", "\[GreaterEqual]", "0"]], "&&", RowBox[List["a", "\[Element]", "Reals"]], "&&", RowBox[List["b", "\[Element]", "Reals"]]]]]]]]]]










Contributed by





A. Grosberg










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





2003-08-21





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