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http://functions.wolfram.com/03.03.06.0048.01
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BesselY[\[Nu], z] \[Proportional]
Piecewise[{{(2/Pi) Log[z/2] + (2/Pi) EulerGamma, \[Nu] == 0},
{-(((-1)^((Abs[\[Nu]] - \[Nu])/2) (Abs[\[Nu]] - 1)!)/Pi)/
(z/2)^Abs[\[Nu]], Element[\[Nu], Integers] && \[Nu] != 0}},
(-((Cos[Pi \[Nu]] Gamma[-\[Nu]])/Pi)) (z/2)^\[Nu] -
Gamma[\[Nu]]/Pi/(z/2)^\[Nu]] /; (z -> 0)
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List["Piecewise", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[RowBox[List[FractionBox["2", "\[Pi]"], " ", RowBox[List["Log", "[", FractionBox["z", "2"], "]"]]]], "+", RowBox[List[FractionBox["2", "\[Pi]"], "EulerGamma"]]]], ",", RowBox[List["\[Nu]", "\[Equal]", "0"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[FractionBox[RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Nu]"]], "2"], " "]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "1"]], ")"]], "!"]]]], "\[Pi]"]]], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]]]], ",", RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "\[And]", RowBox[List["\[Nu]", "!=", "0"]]]]]], "}"]]]], "}"]], ",", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]], "\[Pi]"]]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], "\[Nu]"]]], " ", "-", RowBox[List[FractionBox[RowBox[List["Gamma", "[", "\[Nu]", "]"]], "\[Pi]"], SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["-", "\[Nu]"]]]]]]]]], "]"]]]], "/;", RowBox[List["(", RowBox[List["z", "->", "0"]], ")"]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msub> <mi> Y </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ∝ </mo> <mrow> <mo>  </mo> <mtable> <mtr> <mtd> <mrow> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mtext> </mtext> </mrow> <mi> π </mi> </mfrac> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] </annotation> </semantics> <mo> ⁢ </mo> <mn> 2 </mn> </mrow> <mi> π </mi> </mfrac> </mrow> </mtd> <mtd> <mrow> <mi> ν </mi> <mo>  </mo> <mn> 0 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> π </mi> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mfrac> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> ν </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mn> 2 </mn> </mfrac> <mtext> </mtext> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> ν </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> ν </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> </mrow> </msup> </mrow> </mtd> <mtd> <mrow> <mrow> <mi> ν </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mi> ν </mi> <mo> ≠ </mo> <mn> 0 </mn> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> ν </mi> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mi> π </mi> </mfrac> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> <mi> π </mi> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mi> ν </mi> </msup> </mrow> </mrow> </mtd> <mtd> <semantics> <mi> True </mi> <annotation encoding='Mathematica'> TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]] </annotation> </semantics> </mtd> </mtr> </mtable> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> BesselY </ci> <ci> ν </ci> <ci> z </ci> </apply> <piecewise> <piece> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ln /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <eulergamma /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <eq /> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </piece> <piece> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <abs /> <ci> ν </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <factorial /> <apply> <plus /> <apply> <abs /> <ci> ν </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <abs /> <ci> ν </ci> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> ν </ci> <integers /> </apply> <apply> <neq /> <ci> ν </ci> <cn type='integer'> 0 </cn> </apply> </apply> </piece> <otherwise> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Gamma </ci> <ci> ν </ci> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <cos /> <apply> <times /> <pi /> <ci> ν </ci> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> ν </ci> </apply> </apply> </apply> </apply> </otherwise> </piecewise> </apply> <apply> <ci> Rule </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BesselY", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["\[Piecewise]", GridBox[List[List[RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["Log", "[", FractionBox["z", "2"], "]"]]]], "\[Pi]"], "+", FractionBox[RowBox[List["2", " ", "EulerGamma"]], "\[Pi]"]]], RowBox[List["\[Nu]", "\[Equal]", "0"]]], List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "\[Nu]"]], ")"]]]]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "1"]], ")"]], "!"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]]]], "\[Pi]"]]], RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "&&", RowBox[List["\[Nu]", "\[NotEqual]", "0"]]]]], List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], "\[Nu]"]]], "\[Pi]"]]], "-", FractionBox[RowBox[List[RowBox[List["Gamma", "[", "\[Nu]", "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List["-", "\[Nu]"]]]]], "\[Pi]"]]], TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]]]], Rule[ColumnAlignments, List[Left]], Rule[ColumnSpacings, 1.2`], Rule[ColumnWidths, Automatic]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", "0"]], ")"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
