Bessel function of the second kind: Representations through equivalent functions (formula 03.03.27.0004)

BesselY

$Mathematica Notation$

Bessel-Type Functions

BesselY[nu,z]

Representations through equivalent functions

With related functions

http://functions.wolfram.com/03.03.27.0004.01

Input Form

BesselY[\[Nu], z] == (1/Pi) ((-2 ((-I) z)^\[Nu] z^\[Nu] BesselK[\[Nu], (-I) z] + Pi z^(2 \[Nu]) BesselJ[\[Nu], z] Cot[Pi \[Nu]] - Pi ((-I) z)^(2 \[Nu]) BesselJ[\[Nu], z] Csc[Pi \[Nu]])/z^(2 \[Nu])) /; !Element[\[Nu], Integers]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", "\[Pi]"], RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]], ")"]], "\[Nu]"], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]]], "]"]]]], "+", RowBox[List["\[Pi]", " ", SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], "-", RowBox[List["\[Pi]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]], ")"]], RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]]]], ")"]]]], ")"]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List["\[Nu]", ",", "Integers"]], "]"]], "]"]]]]]]

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> <msub> <mi> Y </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mi> z </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </msup> <mtext> </mtext> </mrow> <mi> π </mi> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> ν </mi> </msup> <mo> ⁢ </mo> <mrow> <msub> <mi> K </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> ν </mi> </msup> </mrow> <mo> - </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <msub> <mi> J </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> csc </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </msup> </mrow> <mo> + </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msub> <mi> J </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cot </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mtext>  </mtext> <mrow> <mi> ν </mi> <mo> ∉ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> BesselY </ci> <ci> ν </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -2 </cn> <ci> ν </ci> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <power /> <ci> z </ci> <ci> ν </ci> </apply> <apply> <ci> BesselK </ci> <ci> ν </ci> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> z </ci> </apply> <ci> ν </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <apply> <ci> BesselJ </ci> <ci> ν </ci> <ci> z </ci> </apply> <apply> <csc /> <apply> <times /> <pi /> <ci> ν </ci> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> </apply> </apply> </apply> <apply> <times /> <pi /> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> </apply> <apply> <ci> BesselJ </ci> <ci> ν </ci> <ci> z </ci> </apply> <apply> <cot /> <apply> <times /> <pi /> <ci> ν </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <notin /> <ci> ν </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[FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], " ", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]], ")"]], "\[Nu]"], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]]], "]"]]]], "+", RowBox[List["\[Pi]", " ", SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["Cot", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], "-", RowBox[List["\[Pi]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]], ")"]], RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], " ", RowBox[List["Csc", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]]]], ")"]]]], "\[Pi]"], "/;", RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]]]

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

2001-10-29