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BesselY






Mathematica Notation

Traditional Notation









Bessel-Type Functions > BesselY[nu,z] > Identities > Recurrence identities > Distant neighbors > Increasing





http://functions.wolfram.com/03.03.17.0014.01









  


  










Input Form





BesselY[\[Nu], z] == (2^(-1 + n) Pochhammer[1 + \[Nu], -1 + n] ((-z) HypergeometricPFQ[{1, (1 - n)/2, 1 - n/2}, {1, 1 - n, 1 - n - \[Nu], 1 + \[Nu]}, -z^2] BesselY[1 + n + \[Nu], z] + 2 (n + \[Nu]) HypergeometricPFQ[{1, (1 - n)/2, -(n/2)}, {1, -n, -n - \[Nu], 1 + \[Nu]}, -z^2] BesselY[n + \[Nu], z]))/z^n /; Element[n, Integers] && n >= 0










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], " ", SuperscriptBox["z", RowBox[List["-", "n"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", "1"]], "+", "n"]]]], "]"]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "z"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", FractionBox[RowBox[List["1", "-", "n"]], "2"], ",", RowBox[List["1", "-", FractionBox["n", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "-", "n"]], ",", RowBox[List["1", "-", "n", "-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]], RowBox[List["BesselY", "[", RowBox[List[RowBox[List["1", "+", "n", "+", "\[Nu]"]], ",", "z"]], "]"]]]], " ", "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["n", "+", "\[Nu]"]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", FractionBox[RowBox[List["1", "-", "n"]], "2"], ",", RowBox[List["-", FractionBox["n", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["-", "n"]], ",", RowBox[List[RowBox[List["-", "n"]], "-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]], " ", RowBox[List["BesselY", "[", RowBox[List[RowBox[List["n", "+", "\[Nu]"]], ",", "z"]], "]"]]]]]], ")"]]]]]], " ", "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]










MathML Form







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</ci> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <apply> <ci> HypergeometricPFQ </ci> <list> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </list> <list> <cn type='integer'> 1 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> </list> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <ci> BesselY </ci> <apply> <plus /> <ci> n </ci> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BesselY", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], " ", SuperscriptBox["z", RowBox[List["-", "n"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", RowBox[List[RowBox[List["-", "1"]], "+", "n"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "z"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", FractionBox[RowBox[List["1", "-", "n"]], "2"], ",", RowBox[List["1", "-", FractionBox["n", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "-", "n"]], ",", RowBox[List["1", "-", "n", "-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]], " ", RowBox[List["BesselY", "[", RowBox[List[RowBox[List["1", "+", "n", "+", "\[Nu]"]], ",", "z"]], "]"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["n", "+", "\[Nu]"]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", FractionBox[RowBox[List["1", "-", "n"]], "2"], ",", RowBox[List["-", FractionBox["n", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["-", "n"]], ",", RowBox[List[RowBox[List["-", "n"]], "-", "\[Nu]"]], ",", RowBox[List["1", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", SuperscriptBox["z", "2"]]]]], "]"]], " ", RowBox[List["BesselY", "[", RowBox[List[RowBox[List["n", "+", "\[Nu]"]], ",", "z"]], "]"]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2007-05-02





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