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BesselY






Mathematica Notation

Traditional Notation









Bessel-Type Functions > BesselY[nu,z] > Integration > Indefinite integration > Involving direct function and Bessel-type functions > Involving Bessel functions > Involving Bessel J and power > Linear arguments





http://functions.wolfram.com/03.03.21.0091.01









  


  










Input Form





Integrate[(1/z) ((-\[Nu]^2 + \[Mu]^2 + (a^2 - b^2) z^2) BesselJ[\[Mu], b z] BesselY[\[Nu], a z]), z] == b z BesselJ[-1 + \[Mu], b z] BesselY[\[Nu], a z] - BesselJ[\[Mu], b z] (a z BesselY[-1 + \[Nu], a z] + (\[Mu] - \[Nu]) BesselY[\[Nu], a z])










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List[FractionBox["1", "z"], RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["\[Nu]", "2"]]], "+", SuperscriptBox["\[Mu]", "2"], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]], ")"]], " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Mu]", ",", RowBox[List["b", " ", "z"]]]], "]"]], " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", RowBox[List["a", " ", "z"]]]], "]"]]]], ")"]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["b", " ", "z", " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Mu]"]], ",", RowBox[List["b", " ", "z"]]]], "]"]], " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", RowBox[List["a", " ", "z"]]]], "]"]]]], "-", RowBox[List[RowBox[List["BesselJ", "[", RowBox[List["\[Mu]", ",", RowBox[List["b", " ", "z"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", "z", " ", RowBox[List["BesselY", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ",", RowBox[List["a", " ", "z"]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]"]], ")"]], " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", RowBox[List["a", " ", "z"]]]], "]"]]]]]], ")"]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mo> &#8747; </mo> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> &#956; </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> &#957; </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> J </mi> <mi> &#956; </mi> </msub> <mo> ( </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> Y </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> z </mi> </mfrac> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <msub> <mi> J </mi> <mrow> <mi> &#956; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> Y </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mrow> <msub> <mi> J </mi> <mi> &#956; </mi> </msub> <mo> ( </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <msub> <mi> Y </mi> <mrow> <mi> &#957; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> &#956; </mi> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> Y </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> &#956; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <ci> BesselJ </ci> <ci> &#956; </ci> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <ci> BesselY </ci> <ci> &#957; </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> b </ci> <ci> z </ci> <apply> <ci> BesselJ </ci> <apply> <plus /> <ci> &#956; </ci> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <ci> BesselY </ci> <ci> &#957; </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> BesselJ </ci> <ci> &#956; </ci> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> a </ci> <ci> z </ci> <apply> <ci> BesselY </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <ci> &#956; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <ci> BesselY </ci> <ci> &#957; </ci> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox["\[Nu]_", "2"]]], "+", SuperscriptBox["\[Mu]_", "2"], "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["a_", "2"], "-", SuperscriptBox["b_", "2"]]], ")"]], " ", SuperscriptBox["z_", "2"]]]]], ")"]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Mu]_", ",", RowBox[List["b_", " ", "z_"]]]], "]"]], " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]_", ",", RowBox[List["a_", " ", "z_"]]]], "]"]]]], "z_"], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["b", " ", "z", " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Mu]"]], ",", RowBox[List["b", " ", "z"]]]], "]"]], " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", RowBox[List["a", " ", "z"]]]], "]"]]]], "-", RowBox[List[RowBox[List["BesselJ", "[", RowBox[List["\[Mu]", ",", RowBox[List["b", " ", "z"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", "z", " ", RowBox[List["BesselY", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ",", RowBox[List["a", " ", "z"]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]"]], ")"]], " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", RowBox[List["a", " ", "z"]]]], "]"]]]]]], ")"]]]]]]]]]]










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





2001-10-29