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BesselY






Mathematica Notation

Traditional Notation









Bessel-Type Functions > BesselY[nu,z] > Representations through more general functions > Through Meijer G > Classical cases involving 0F1





http://functions.wolfram.com/03.03.26.0117.01









  


  










Input Form





BesselY[\[Nu], 2 Sqrt[z]] Hypergeometric0F1[-n + \[Nu], -z] == ((2^(-1 - n) Gamma[-n + \[Nu]])/(z^(\[Nu]/2) Sqrt[Pi])) (2^\[Nu] z^(\[Nu]/2) MeijerG[{{(1/2) (1 + n - \[Nu]), (1/2) (2 + n - \[Nu])}, {1/2 + n - \[Nu]/2}}, {{1 + n - \[Nu]/2, \[Nu]/2}, {1 + n - (3 \[Nu])/2, -(\[Nu]/2), 1/2 + n - \[Nu]/2}}, 4 z] - 2 Csc[\[Nu] Pi] Sum[(((-1)^(k + Floor[(1 + n)/2]) 4^k z^k Gamma[1/2 + k - n + Floor[n/2]])/(k! Gamma[1 + k - \[Nu]] Gamma[k - n + \[Nu]])) Pochhammer[1 - k + Floor[n/2], n - Floor[n/2]], {k, 0, Floor[n/2]}]) /; Element[n, Integers] && n >= 0










Standard Form





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MathML Form







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</ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <ci> &#957; </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </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[RowBox[List["BesselY", "[", RowBox[List["\[Nu]_", ",", RowBox[List["2", " ", SqrtBox["z_"]]]]], "]"]], " ", RowBox[List["Hypergeometric0F1", "[", RowBox[List[RowBox[List[RowBox[List["-", "n_"]], "+", "\[Nu]_"]], ",", RowBox[List["-", "z_"]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", SuperscriptBox["z", RowBox[List["-", FractionBox["\[Nu]", "2"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "n"]], "+", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", SuperscriptBox["z", RowBox[List["\[Nu]", "/", "2"]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "n", "-", "\[Nu]"]], ")"]]]], ",", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "n", "-", "\[Nu]"]], ")"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], "+", "n", "-", FractionBox["\[Nu]", "2"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "+", "n", "-", FractionBox["\[Nu]", "2"]]], ",", FractionBox["\[Nu]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "n", "-", FractionBox[RowBox[List["3", " ", "\[Nu]"]], "2"]]], ",", RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", "2"], "+", "n", "-", FractionBox["\[Nu]", "2"]]]]], "}"]]]], "}"]], ",", RowBox[List["4", " ", "z"]]]], "]"]]]], "-", RowBox[List["2", " ", RowBox[List["Csc", "[", RowBox[List["\[Nu]", " ", "\[Pi]"]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["1", "+", "n"]], "2"], "]"]]]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox["z", "k"], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k", "-", "n", "+", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], "]"]]]], ")"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", "k", "+", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], ",", RowBox[List["n", "-", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "k", "-", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["k", "-", "n", "+", "\[Nu]"]], "]"]]]]]]]]]]], ")"]]]], SqrtBox["\[Pi]"]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2007-05-02