Bessel function of the second kind: Representations through more general functions (formula 03.03.26.0124)

BesselY

$Mathematica Notation$

Bessel-Type Functions

BesselY[nu,z]

Representations through more general functions

Through Meijer G

Classical cases involving ₀F₁

http://functions.wolfram.com/03.03.26.0124.01

Input Form

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

Standard Form

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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> ν </mi> </msub> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 0 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mo>   </mo> <mo> ; </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "0"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox["\[Null]", InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["\[Nu]", "+", "1"]], Hypergeometric0F1, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric0F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox["z", Hypergeometric0F1, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], Hypergeometric0F1] </annotation> </semantics> </mrow> <mo>  </mo> <mrow> <mrow> <mo> - </mo> <msup> <mn> 2 </mn> <mrow> <mo> - </mo> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> </mrow> </msup> </mrow> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <semantics> <mrow> <msubsup> <mi> G </mi> <mrow> <mn> 2 </mn> <mo> , </mo> <mn> 6 </mn> </mrow> <mrow> <mn> 3 </mn> <mo> , </mo> <mn> 0 </mn> </mrow> </msubsup> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <mrow> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> <mo> - </mo> <mfrac> <mi> ν </mi> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> - </mo> <mfrac> <mi> ν </mi> <mn> 4 </mn> </mfrac> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mfrac> <mi> ν </mi> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mi> ν </mi> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <mi> ν </mi> <mn> 4 </mn> </mfrac> <mo> , </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> <mo> - </mo> <mfrac> <mi> ν </mi> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> - </mo> <mfrac> <mi> ν </mi> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[SubsuperscriptBox[TagBox["G", MeijerG], RowBox[List["2", ",", "6"]], RowBox[List["3", ",", "0"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[FractionBox[SuperscriptBox["z", "2"], "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], "\[VerticalSeparator]", GridBox[List[List[RowBox[List[TagBox[RowBox[List[FractionBox["3", "4"], "-", FractionBox["\[Nu]", "4"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["1", "4"], "-", FractionBox["\[Nu]", "4"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]], List[RowBox[List[TagBox[RowBox[List[FractionBox["1", "2"], "-", FractionBox["\[Nu]", "4"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["-", FractionBox["\[Nu]", "4"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["\[Nu]", "4"], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["3", "4"], "-", FractionBox["\[Nu]", "4"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[FractionBox["1", "4"], "-", FractionBox["\[Nu]", "4"]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["3", " ", "\[Nu]"]], ")"]]]], MeijerG, Rule[Editable, True], Rule[Selectable, True]]]]]]]]], ")"]]]], MeijerG, Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> </mrow> </mrow> <mo> /; 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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["1", "+", "\[Nu]_"]], ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["2", RowBox[List["-", FractionBox["\[Nu]", "2"]]]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["Gamma", "[", RowBox[List["1", "+", "\[Nu]"]], "]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "4"], "-", FractionBox["\[Nu]", "4"]]], ",", RowBox[List[FractionBox["1", "4"], "-", FractionBox["\[Nu]", "4"]]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "-", FractionBox["\[Nu]", "4"]]], ",", RowBox[List["-", FractionBox["\[Nu]", "4"]]], ",", FractionBox["\[Nu]", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "4"], "-", FractionBox["\[Nu]", "4"]]], ",", RowBox[List[FractionBox["1", "4"], "-", FractionBox["\[Nu]", "4"]]], ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["3", " ", "\[Nu]"]], ")"]]]]]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "4"]]], "]"]]]], "/;", RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "2"]]], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", FractionBox["\[Pi]", "2"]]]]]]]]]

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

2007-05-02