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BesselY




Mathematica Notation

Traditional Notation









Bessel-Type Functions > BesselY[nu,z] > Representations through more general functions > Through Meijer G > Generalized cases for products of Y





http://functions.wolfram.com/03.03.26.0051.01









  


  








Input Form





BesselY[\[Mu], z] BesselY[\[Nu], z] == (2/Sqrt[Pi]) MeijerG[{{}, {0, 1/2, (1 - \[Mu] - \[Nu])/2}}, {{(\[Mu] + \[Nu])/2, (\[Mu] - \[Nu])/2, (\[Nu] - \[Mu])/2, -((\[Mu] + \[Nu])/2)}, {(1 - \[Mu] - \[Nu])/2}}, z, 1/2] + (1/Sqrt[Pi]) MeijerG[{{0, 1/2}, {}}, {{(\[Mu] + \[Nu])/2}, {(\[Mu] - \[Nu])/2, (\[Nu] - \[Mu])/2, -((\[Mu] + \[Nu])/2)}}, z, 1/2] /; !(Element[-\[Mu] - \[Nu] - 1, Integers] && -\[Mu] - \[Nu] - 1 >= 0)








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["BesselY", "[", RowBox[List["\[Mu]", ",", "z"]], "]"]], " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["2", SqrtBox["\[Pi]"]], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"], ",", FractionBox[RowBox[List["1", "-", "\[Mu]", "-", "\[Nu]"]], "2"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[Mu]", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["\[Mu]", "-", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["\[Nu]", "-", "\[Mu]"]], "2"], ",", RowBox[List["-", FractionBox[RowBox[List["\[Mu]", "+", "\[Nu]"]], "2"]]]]], "}"]], ",", RowBox[List["{", FractionBox[RowBox[List["1", "-", "\[Mu]", "-", "\[Nu]"]], "2"], "}"]]]], "}"]], ",", "z", ",", FractionBox["1", "2"]]], "]"]]]], "+", RowBox[List[FractionBox["1", SqrtBox["\[Pi]"]], RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox[RowBox[List["\[Mu]", "+", "\[Nu]"]], "2"], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[Mu]", "-", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["\[Nu]", "-", "\[Mu]"]], "2"], ",", RowBox[List["-", FractionBox[RowBox[List["\[Mu]", "+", "\[Nu]"]], "2"]]]]], "}"]]]], "}"]], ",", "z", ",", FractionBox["1", "2"]]], "]"]]]]]]]], "/;", RowBox[List["Not", "[", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]", "-", "1"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]", "-", "1"]], "\[GreaterEqual]", "0"]]]], "]"]]]]]]








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> <mrow> <msub> <mi> Y </mi> <mi> &#956; </mi> </msub> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> Y </mi> <mi> &#957; </mi> </msub> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mtext> </mtext> </mrow> <msqrt> <mi> &#960; </mi> </msqrt> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 3 </mn> <mo> , </mo> <mn> 5 </mn> </mrow> <mrow> <mn> 4 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> </msubsup> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> &#10072; </mo> <mtable> <mtr> <mtd> <mrow> <mn> 0 </mn> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#956; </mi> <mo> - </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mi> &#956; </mi> <mo> + </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> &#956; </mi> <mo> - </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> &#957; </mi> <mo> - </mo> <mi> &#956; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> &#956; </mi> <mo> + </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> &#956; </mi> <mo> - </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox[&quot;G&quot;, MeijerG], RowBox[List[&quot;3&quot;, &quot;,&quot;, &quot;5&quot;]], RowBox[List[&quot;4&quot;, &quot;,&quot;, &quot;0&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[RowBox[List[TagBox[&quot;z&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;1&quot;, &quot;2&quot;], MeijerG, Rule[Editable, True]]]], MeijerG], &quot;\[VerticalSeparator]&quot;, GridBox[List[List[RowBox[List[TagBox[&quot;0&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;1&quot;, &quot;2&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;\[Mu]&quot;, &quot;-&quot;, &quot;\[Nu]&quot;]], &quot;2&quot;], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[FractionBox[RowBox[List[&quot;\[Mu]&quot;, &quot;+&quot;, &quot;\[Nu]&quot;]], &quot;2&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[&quot;\[Mu]&quot;, &quot;-&quot;, &quot;\[Nu]&quot;]], &quot;2&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[&quot;\[Nu]&quot;, &quot;-&quot;, &quot;\[Mu]&quot;]], &quot;2&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;-&quot;, FractionBox[RowBox[List[&quot;\[Mu]&quot;, &quot;+&quot;, &quot;\[Nu]&quot;]], &quot;2&quot;]]], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;\[Mu]&quot;, &quot;-&quot;, &quot;\[Nu]&quot;]], &quot;2&quot;], MeijerG, Rule[Editable, True]]]]]]]]], &quot;)&quot;]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mi> &#960; </mi> </msqrt> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 4 </mn> </mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> </mrow> </msubsup> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> &#10072; </mo> <mtable> <mtr> <mtd> <mrow> <mn> 0 </mn> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mi> &#956; </mi> <mo> + </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> &#956; </mi> <mo> - </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> &#957; </mi> <mo> - </mo> <mi> &#956; </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> &#956; </mi> <mo> + </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox[&quot;G&quot;, MeijerG], RowBox[List[&quot;2&quot;, &quot;,&quot;, &quot;4&quot;]], RowBox[List[&quot;1&quot;, &quot;,&quot;, &quot;2&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[RowBox[List[TagBox[&quot;z&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;1&quot;, &quot;2&quot;], MeijerG, Rule[Editable, True]]]], MeijerG], &quot;\[VerticalSeparator]&quot;, GridBox[List[List[RowBox[List[TagBox[&quot;0&quot;, MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;1&quot;, &quot;2&quot;], MeijerG, Rule[Editable, True]]]]], List[RowBox[List[TagBox[FractionBox[RowBox[List[&quot;\[Mu]&quot;, &quot;+&quot;, &quot;\[Nu]&quot;]], &quot;2&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[&quot;\[Mu]&quot;, &quot;-&quot;, &quot;\[Nu]&quot;]], &quot;2&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[RowBox[List[&quot;\[Nu]&quot;, &quot;-&quot;, &quot;\[Mu]&quot;]], &quot;2&quot;], MeijerG, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[&quot;-&quot;, FractionBox[RowBox[List[&quot;\[Mu]&quot;, &quot;+&quot;, &quot;\[Nu]&quot;]], &quot;2&quot;]]], MeijerG, Rule[Editable, True]]]]]]]]], &quot;)&quot;]]]], MeijerG, Rule[Editable, False]] </annotation> </semantics> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> &#956; </mi> </mrow> <mo> - </mo> <mi> &#957; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> &#8713; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> Y </ci> <ci> &#956; </ci> </apply> <ci> z </ci> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> Y </ci> <ci> &#957; </ci> </apply> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> MeijerG </ci> <list> <list /> <list> <cn type='integer'> 0 </cn> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; 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</ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> &#956; </ci> <ci> &#957; </ci> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </list> </list> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <notin /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#956; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> -1 </cn> </apply> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["BesselY", "[", RowBox[List["\[Mu]_", ",", "z_"]], "]"]], " ", RowBox[List["BesselY", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "\[Mu]", "-", "\[Nu]"]], ")"]]]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[Mu]", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["\[Mu]", "-", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["\[Nu]", "-", "\[Mu]"]], "2"], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]"]], ")"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", "\[Mu]", "-", "\[Nu]"]], ")"]]]], "}"]]]], "}"]], ",", "z", ",", FractionBox["1", "2"]]], "]"]]]], SqrtBox["\[Pi]"]], "+", FractionBox[RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "2"]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", FractionBox[RowBox[List["\[Mu]", "+", "\[Nu]"]], "2"], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[Mu]", "-", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["\[Nu]", "-", "\[Mu]"]], "2"], ",", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]"]], ")"]]]]]], "}"]]]], "}"]], ",", "z", ",", FractionBox["1", "2"]]], "]"]], SqrtBox["\[Pi]"]]]], "/;", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]", "-", "1"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List["-", "\[Mu]"]], "-", "\[Nu]", "-", "1"]], "\[GreaterEqual]", "0"]]]], ")"]]]]]]]]]]


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





2001-10-29