Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











BesselY






Mathematica Notation

Traditional Notation









Bessel-Type Functions > BesselY[nu,z] > Series representations > Asymptotic series expansions > Expansions for any z in exponential form > Using exponential function with branch cut-free arguments





http://functions.wolfram.com/03.03.06.0068.01









  


  










Input Form





BesselY[\[Nu], z] \[Proportional] (((-1)^(1/4) z^(-1 + \[Nu]) (z^2)^(-(1/4) - \[Nu]/2))/(2 Sqrt[2 Pi])) (((I E^((I Pi \[Nu])/2) (I + E^(I Pi \[Nu])) z - I E^((I Pi \[Nu])/2) (-I + E^(I Pi \[Nu])) Sqrt[z^2] - 2 I E^((I Pi \[Nu])/2) (-I + E^(I Pi \[Nu])) z Floor[1/2 - Arg[z]/Pi] + 2 I E^((I Pi \[Nu])/2) (I + E^(I Pi \[Nu])) Sqrt[z^2] Floor[1/2 - Arg[z]/Pi]) HypergeometricPFQ[{1/2 + \[Nu], 1/2 - \[Nu]}, {}, -(I/(2 z))])/E^(I (Pi \[Nu] - z)) - ((E^((I Pi \[Nu])/2) (1 - I E^(I Pi \[Nu])) z - E^((I Pi \[Nu])/2) (1 + I E^(I Pi \[Nu])) Sqrt[z^2] + 2 E^((I Pi \[Nu])/2) (1 + I E^(I Pi \[Nu])) z Floor[1/2 - Arg[z]/Pi] + 2 I E^((I Pi \[Nu])/2) (I + E^(I Pi \[Nu])) Sqrt[z^2] Floor[1/2 - Arg[z]/Pi]) HypergeometricPFQ[{1/2 + \[Nu], 1/2 - \[Nu]}, {}, I/(2 z)])/E^(I (Pi \[Nu] + z))) /; (Abs[z] -> Infinity) && !Element[\[Nu], Integers]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], "-", FractionBox["\[Nu]", "2"]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", "\[Nu]"]], "-", "z"]], ")"]]]]], RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]], ")"]], " ", "z"]], "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]], ")"]], " ", SqrtBox[SuperscriptBox["z", "2"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]], ")"]], " ", "z", " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", "z", "]"]], "\[Pi]"]]], "]"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]], ")"]], " ", SqrtBox[SuperscriptBox["z", "2"]], " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", "z", "]"]], "\[Pi]"]]], "]"]]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["-", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "z"]]]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", "\[Nu]"]], "+", "z"]], ")"]]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]]]], ")"]], " ", "z"]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]]]], ")"]], " ", SqrtBox[SuperscriptBox["z", "2"]]]], "+", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]]]], ")"]], " ", "z", " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", "z", "]"]], "\[Pi]"]]], "]"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]], ")"]], " ", SqrtBox[SuperscriptBox["z", "2"]], " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", "z", "]"]], "\[Pi]"]]], "]"]]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "z"]]]]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", 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> &#8733; </mo> <mrow> <mfrac> <mrow> <mroot> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </mroot> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mi> &#957; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mi> &#957; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> </msup> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> + </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> + </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> + </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mi> &#960; </mi> </mfrac> </mrow> <mo> &#8971; </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> + </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mi> &#960; </mi> </mfrac> </mrow> <mo> &#8971; </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 0 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> &#957; </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> &#957; </mi> </mrow> </mrow> <mo> ; </mo> <mo> &#8202; </mo> <mo> ; </mo> <mrow> <mo> - </mo> <mfrac> <mi> &#8520; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, &quot;2&quot;], SubscriptBox[&quot;F&quot;, &quot;0&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[&quot;\[Nu]&quot;, &quot;+&quot;, FractionBox[&quot;1&quot;, &quot;2&quot;]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;1&quot;, &quot;2&quot;], &quot;-&quot;, &quot;\[Nu]&quot;]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], &quot;;&quot;, TagBox[TagBox[&quot;\[Null]&quot;, InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], &quot;;&quot;, TagBox[RowBox[List[&quot;-&quot;, FractionBox[&quot;\[ImaginaryI]&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;z&quot;]]]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQ] </annotation> </semantics> </mrow> <mo> - </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mi> &#960; </mi> </mfrac> </mrow> <mo> &#8971; </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> &#8290; </mo> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> + </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mi> &#960; </mi> </mfrac> </mrow> <mo> &#8971; </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 0 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> &#957; </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> &#957; </mi> </mrow> </mrow> <mo> ; </mo> <mo> &#8202; </mo> <mo> ; </mo> <mfrac> <mi> &#8520; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, &quot;2&quot;], SubscriptBox[&quot;F&quot;, &quot;0&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List[&quot;\[Nu]&quot;, &quot;+&quot;, FractionBox[&quot;1&quot;, &quot;2&quot;]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;1&quot;, &quot;2&quot;], &quot;-&quot;, &quot;\[Nu]&quot;]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], &quot;;&quot;, TagBox[TagBox[&quot;\[Null]&quot;, InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], &quot;;&quot;, TagBox[FractionBox[&quot;\[ImaginaryI]&quot;, RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;z&quot;]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mi> &#957; </mi> <mo> &#8713; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> BesselY </ci> <ci> &#957; </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <plus /> <apply> <times /> <pi /> <ci> &#957; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <imaginaryi /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> &#957; </ci> </apply> </apply> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> &#957; </ci> </apply> </apply> </apply> <ci> z </ci> <apply> <floor /> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <arg /> <ci> z </ci> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <imaginaryi /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <floor /> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <arg /> <ci> z </ci> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <plus /> <ci> &#957; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </list> <list /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <pi /> <ci> &#957; </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> z </ci> <apply> <floor /> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <arg /> <ci> z </ci> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> &#957; </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> &#957; </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> &#957; </ci> </apply> </apply> </apply> </apply> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <imaginaryi /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> &#957; </ci> </apply> </apply> </apply> <imaginaryi /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <floor /> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <arg /> <ci> z </ci> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <plus /> <ci> &#957; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> </list> <list /> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <apply> <abs /> <ci> z </ci> </apply> <infinity /> </apply> <apply> <notin /> <ci> &#957; </ci> <integers /> </apply> </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[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], "-", FractionBox["\[Nu]", "2"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", "\[Nu]"]], "-", "z"]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]], ")"]], " ", "z"]], "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]], ")"]], " ", SqrtBox[SuperscriptBox["z", "2"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]], ")"]], " ", "z", " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", "z", "]"]], "\[Pi]"]]], "]"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]], ")"]], " ", SqrtBox[SuperscriptBox["z", "2"]], " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", "z", "]"]], "\[Pi]"]]], "]"]]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["-", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "z"]]]]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", "\[Nu]"]], "+", "z"]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]]]], ")"]], " ", "z"]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]]]], ")"]], " ", SqrtBox[SuperscriptBox["z", "2"]]]], "+", RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]]]], ")"]], " ", "z", " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", "z", "]"]], "\[Pi]"]]], "]"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]], ")"]], " ", SqrtBox[SuperscriptBox["z", "2"]], " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", "z", "]"]], "\[Pi]"]]], "]"]]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", "z"]]]]], "]"]]]]]], ")"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "&&", RowBox[List["!", RowBox[List["\[Nu]", "\[Element]", "Integers"]]]]]]]]]]]]










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





2007-05-02





© 1998-2014 Wolfram Research, Inc.