Bessel function of the second kind: Representations through more general functions (formula 03.03.26.0165)

BesselY

$Mathematica Notation$

Bessel-Type Functions

BesselY[nu,z]

Representations through more general functions

Through Meijer G

Generalized cases involving ₀F₁

http://functions.wolfram.com/03.03.26.0165.01

Input Form

BesselY[\[Nu], z] Hypergeometric0F1[-\[Nu], -(z^2/4)] == 2^(-1 - \[Nu]) ((2 z^\[Nu] Cot[Pi \[Nu]])/Gamma[1 + \[Nu]] + (Gamma[-\[Nu]]/Sqrt[Pi]) MeijerG[{{(1 + \[Nu])/2, (2 + \[Nu])/2}, {(1 - \[Nu])/2}}, {{(2 + \[Nu])/2, -(\[Nu]/2)}, {1 + (3 \[Nu])/2, \[Nu]/2, (1 - \[Nu])/2}}, z, 1/2])

Standard Form

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MathML Form

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Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["BesselY", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], " ", RowBox[List["Hypergeometric0F1", "[", RowBox[List[RowBox[List["-", "\[Nu]_"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z_", "2"], "4"]]]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]]], "+", FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "+", "\[Nu]"]], "2"], ",", FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"]]], "}"]], ",", RowBox[List["{", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["2", "+", "\[Nu]"]], "2"], ",", RowBox[List["-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", FractionBox[RowBox[List["3", " ", "\[Nu]"]], "2"]]], ",", FractionBox["\[Nu]", "2"], ",", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"]]], "}"]]]], "}"]], ",", "z", ",", FractionBox["1", "2"]]], "]"]]]], SqrtBox["\[Pi]"]]]], ")"]]]]]]]]

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

2007-05-02