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BesselY






Mathematica Notation

Traditional Notation









Bessel-Type Functions > BesselY[nu,z] > Representations through equivalent functions > With related functions





http://functions.wolfram.com/03.03.27.0003.02









  


  










Input Form





BesselY[\[Nu], z] == ((1/Pi) (-2 (I z)^(2 \[Nu]) BesselK[\[Nu], I z] + Pi BesselI[\[Nu], I z] (-(I z)^(2 \[Nu]) + z^(2 \[Nu]) Cos[Pi \[Nu]]) Csc[Pi \[Nu]]))/((I z)^\[Nu] z^\[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]"], SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "z"]], ")"]], RowBox[List["-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "z"]], ")"]], RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]], "+", RowBox[List["\[Pi]", " ", RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "z"]], ")"]], RowBox[List["2", " ", "\[Nu]"]]]]], "+", RowBox[List[SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]]]], ")"]], " ", 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> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> </msup> <mtext> </mtext> </mrow> <mi> &#960; </mi> </mfrac> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mi> csc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> I </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> K </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mtext> &#8203;&#8203; </mtext> <mrow> <mi> &#957; </mi> <mo> &#8713; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> BesselY </ci> <ci> &#957; </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <pi /> <apply> <csc /> <apply> <times /> <pi /> <ci> &#957; </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <cos /> <apply> <times /> <pi /> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> </apply> </apply> </apply> <apply> <ci> BesselI </ci> <ci> &#957; </ci> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <ci> BesselK </ci> <ci> &#957; </ci> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <notin /> <ci> &#957; </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[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "z"]], ")"]], RowBox[List["-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List["-", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "z"]], ")"]], RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]], "+", RowBox[List["\[Pi]", " ", RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "z"]], ")"]], RowBox[List["2", " ", "\[Nu]"]]]]], "+", RowBox[List[SuperscriptBox["z", RowBox[List["2", " ", "\[Nu]"]]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]]]], ")"]], " ", 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