Bessel function of the second kind: Series representations (formula 03.03.06.0074)

BesselY

$Mathematica Notation$

Bessel-Type Functions

BesselY[nu,z]

Series representations

Asymptotic series expansions

Expansions for any z in trigonometric form

Using trigonometric functions with branch cut-free arguments

http://functions.wolfram.com/03.03.06.0074.01

Input Form

BesselY[\[Nu], z] \[Proportional] (-((1 + I)/(E^((1/2) I (Pi \[Nu] + 2 z)) (2 Pi^(1/2) (-z)^(1/2))))) (E^(2 I z) ((2 Sqrt[-z^2])/z - I) - I (((-1)^\[Nu] Sqrt[-z^2])/z)) (1 - ((1 - 2 \[Nu]) (3 - 2 \[Nu]) (1 + 2 \[Nu]) (3 + 2 \[Nu]))/ (128 z^2) + (11025 - 51664 \[Nu]^2 + 31584 \[Nu]^4 - 5376 \[Nu]^6 + 256 \[Nu]^8)/(98304 z^4) + \[Ellipsis]) + (((1 - I) (1 - 4 \[Nu]^2))/(E^((1/2) I (Pi \[Nu] + 2 z)) (16 Pi^(1/2) (-z)^(3/2)))) (E^(2 I z) ((2 Sqrt[-z^2])/z - I) + I (-1)^\[Nu] (Sqrt[-z^2]/z)) (1 - ((3 - 2 \[Nu]) (5 - 2 \[Nu]) (3 + 2 \[Nu]) (5 + 2 \[Nu]))/ (384 z^2) + (893025 - 656784 \[Nu]^2 + 137824 \[Nu]^4 - 10496 \[Nu]^6 + 256 \[Nu]^8)/(491520 z^4) + \[Ellipsis]) /; (Abs[z] -> Infinity) && Element[\[Nu], Integers]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BesselY", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[ImaginaryI]"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", "\[Nu]"]], "+", RowBox[List["2", " ", "z"]]]], ")"]]]]]]], RowBox[List["2", SuperscriptBox["\[Pi]", RowBox[List["1", "/", "2"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["1", "/", "2"]]]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]], "z"], "-", "\[ImaginaryI]"]], " ", ")"]]]], "-", RowBox[List["\[ImaginaryI]", " ", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "\[Nu]"], SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]], "z"]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], RowBox[List["128", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List["11025", "-", RowBox[List["51664", " ", SuperscriptBox["\[Nu]", "2"]]], "+", RowBox[List["31584", " ", SuperscriptBox["\[Nu]", "4"]]], "-", RowBox[List["5376", " ", SuperscriptBox["\[Nu]", "6"]]], "+", RowBox[List["256", " ", SuperscriptBox["\[Nu]", "8"]]]]], RowBox[List["98304", " ", SuperscriptBox["z", "4"]]]], "+", "\[Ellipsis]"]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", "\[ImaginaryI]"]], ")"]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", "\[Nu]"]], "+", RowBox[List["2", " ", "z"]]]], ")"]]]]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["4", " ", SuperscriptBox["\[Nu]", "2"]]]]], ")"]]]], RowBox[List["16", SuperscriptBox["\[Pi]", RowBox[List["1", "/", "2"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "z"]], ")"]], RowBox[List["3", "/", "2"]]]]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]]], " ", RowBox[List["(", " ", RowBox[List[FractionBox[RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]], "z"], "-", "\[ImaginaryI]"]], ")"]]]], "+", RowBox[List["\[ImaginaryI]", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "\[Nu]"], " ", FractionBox[SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], "z"]]]]], ")"]], RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["5", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["5", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], RowBox[List["384", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List["893025", "-", RowBox[List["656784", " ", SuperscriptBox["\[Nu]", "2"]]], "+", RowBox[List["137824", " ", SuperscriptBox["\[Nu]", "4"]]], "-", RowBox[List["10496", " ", SuperscriptBox["\[Nu]", "6"]]], "+", RowBox[List["256", " ", SuperscriptBox["\[Nu]", "8"]]]]], RowBox[List["491520", " ", SuperscriptBox["z", "4"]]]], "+", "\[Ellipsis]"]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]], "\[And]", RowBox[List["\[Nu]", "\[Element]", "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> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> ν </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 16 </mn> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <mi> z </mi> </mfrac> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> ν </mi> </msup> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <mi> z </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> + </mo> <mn> 5 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 5 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 384 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mrow> <mn> 256 </mn> <mo> ⁢ </mo> <msup> <mi> ν </mi> <mn> 8 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 10496 </mn> <mo> ⁢ </mo> <msup> <mi> ν </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 137824 </mn> <mo> ⁢ </mo> <msup> <mi> ν </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 656784 </mn> <mo> ⁢ </mo> <msup> <mi> ν </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 893025 </mn> </mrow> <mrow> <mn> 491520 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mo> … </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mi> ⅈ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <mi> z </mi> </mfrac> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> ν </mi> </msup> <mo> ⁢ </mo> <msqrt> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <mi> z </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> + </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 128 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mrow> <mn> 256 </mn> <mo> ⁢ </mo> <msup> <mi> ν </mi> <mn> 8 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 5376 </mn> <mo> ⁢ </mo> <msup> <mi> ν </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 31584 </mn> <mo> ⁢ </mo> <msup> <mi> ν </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 51664 </mn> <mo> ⁢ </mo> <msup> <mi> ν </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 11025 </mn> </mrow> <mrow> <mn> 98304 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mo> … </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> z </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ∧ </mo> <mrow> <mi> ν </mi> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> </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> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='complex-cartesian'> 1 <sep /> -1 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <imaginaryi /> <apply> <plus /> <apply> <times /> <pi /> <ci> ν </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> ν </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <imaginaryi /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> <cn type='integer'> 5 </cn> </apply> <apply> <plus /> <cn type='integer'> 5 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 384 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 256 </cn> <apply> <power /> <ci> ν </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 10496 </cn> <apply> <power /> <ci> ν </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 137824 </cn> <apply> <power /> <ci> ν </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 656784 </cn> <apply> <power /> <ci> ν </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 893025 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 491520 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> … </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <imaginaryi /> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <imaginaryi /> <apply> <plus /> <apply> <times /> <pi /> <ci> ν </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </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 /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> ν </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 128 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 256 </cn> <apply> <power /> <ci> ν </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 5376 </cn> <apply> <power /> <ci> ν </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 31584 </cn> <apply> <power /> <ci> ν </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 51664 </cn> <apply> <power /> <ci> ν </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 11025 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 98304 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> … </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <apply> <abs /> <ci> z </ci> </apply> <infinity /> </apply> <apply> <in /> <ci> ν </ci> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2007-05-02