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http://functions.wolfram.com/03.03.21.0079.01
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Integrate[BesselJ[-\[Nu], a Sqrt[z]] BesselY[\[Nu], b Sqrt[z]], z] ==
(1/(a^2 - b^2)) (2 Sqrt[z] (a BesselJ[1 - \[Nu], a Sqrt[z]]
BesselY[\[Nu], b Sqrt[z]] + b BesselJ[-\[Nu], a Sqrt[z]]
(BesselJ[1 - \[Nu], b Sqrt[z]] + BesselJ[-1 + \[Nu], b Sqrt[z]]
Cos[Pi \[Nu]]) Csc[Pi \[Nu]]))
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["a", " ", SqrtBox[RowBox[List[" ", "z"]]]]]]], "]"]], RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", RowBox[List["b", " ", SqrtBox[RowBox[List[" ", "z"]]]]]]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]], RowBox[List["(", RowBox[List["2", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["a", " ", SqrtBox["z"]]]]], "]"]], " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", RowBox[List["b", " ", SqrtBox["z"]]]]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["a", " ", SqrtBox["z"]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["b", " ", SqrtBox["z"]]]]], "]"]], "+", RowBox[List[RowBox[List["BesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ",", RowBox[List["b", " ", SqrtBox["z"]]]]], "]"]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]]]], ")"]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]]]], ")"]]]], ")"]]]]]]]]
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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> <mo> ∫ </mo> <mrow> <mrow> <mrow> <msub> <mi> J </mi> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> Y </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> <mtext> </mtext> </mrow> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <msub> <mi> J </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> ν </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> Y </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <msub> <mi> J </mi> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> J </mi> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> ν </mi> </mrow> </msub> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mrow> <msub> <mi> J </mi> <mrow> <mi> ν </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> csc </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <ci> BesselJ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <ci> BesselY </ci> <ci> ν </ci> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <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> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <ci> BesselJ </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <ci> BesselY </ci> <ci> ν </ci> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <ci> BesselJ </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <ci> BesselJ </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <ci> BesselJ </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <cos /> <apply> <times /> <pi /> <ci> ν </ci> </apply> </apply> </apply> </apply> <apply> <csc /> <apply> <times /> <pi /> <ci> ν </ci> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["-", "\[Nu]_"]], ",", RowBox[List["a_", " ", SqrtBox["z_"]]]]], "]"]], " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]_", ",", RowBox[List["b_", " ", SqrtBox["z_"]]]]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["2", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List["a", " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["a", " ", SqrtBox["z"]]]]], "]"]], " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", RowBox[List["b", " ", SqrtBox["z"]]]]], "]"]]]], "+", RowBox[List["b", " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["a", " ", SqrtBox["z"]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["1", "-", "\[Nu]"]], ",", RowBox[List["b", " ", SqrtBox["z"]]]]], "]"]], "+", RowBox[List[RowBox[List["BesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ",", RowBox[List["b", " ", SqrtBox["z"]]]]], "]"]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]]]], ")"]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]]]], ")"]]]], RowBox[List[SuperscriptBox["a", "2"], "-", SuperscriptBox["b", "2"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
