Kelvin function of the second kind: Series representations (formula 03.16.06.0018)

KelvinKer

$Mathematica Notation$

Bessel-Type Functions

KelvinKer[z]

Series representations

Asymptotic series expansions

Expansions inside Stokes sectors

Expansions containing z->infinity

In trigonometric form ||| In trigonometric form

http://functions.wolfram.com/03.16.06.0018.01

Input Form

KelvinKer[\[Nu], z] \[Proportional] (Sqrt[Pi]/(E^(z/Sqrt[2]) Sqrt[2 z])) (Cos[(1/8) (4 Sqrt[2] z + Pi (1 + 4 \[Nu]))] - ((1 - 4 \[Nu]^2)/(8 z)) Sin[(1/8) (-4 Sqrt[2] z + Pi (1 - 4 \[Nu]))] + ((9 - 40 \[Nu]^2 + 16 \[Nu]^4)/(128 z^2)) Sin[(1/8) (-4 Sqrt[2] z - Pi (1 + 4 \[Nu]))] + ((225 - 1036 \[Nu]^2 + 560 \[Nu]^4 - 64 \[Nu]^6)/(3072 z^3)) Cos[(1/8) (4 Sqrt[2] z - Pi (1 - 4 \[Nu]))] + \[Ellipsis]) /; (Abs[z] -> Infinity)

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinKer", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[FractionBox[RowBox[List[SqrtBox["\[Pi]"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]]]], RowBox[List[SqrtBox[RowBox[List["2", "z"]]], " "]]], RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SqrtBox["2"], " ", "z"]], "+", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]]]], ")"]]]], "]"]], "-", RowBox[List[FractionBox[RowBox[List["1", "-", RowBox[List["4", " ", SuperscriptBox["\[Nu]", "2"]]]]], RowBox[List["8", " ", "z"]]], RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SqrtBox["2"], " ", "z"]], "+", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]]]], ")"]]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List["9", "-", RowBox[List["40", " ", SuperscriptBox["\[Nu]", "2"]]], "+", RowBox[List["16", " ", SuperscriptBox["\[Nu]", "4"]]]]], RowBox[List["128", " ", SuperscriptBox["z", "2"]]]], RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SqrtBox["2"], " ", "z"]], "-", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]]]], ")"]]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List["225", "-", RowBox[List["1036", " ", SuperscriptBox["\[Nu]", "2"]]], "+", RowBox[List["560", " ", SuperscriptBox["\[Nu]", "4"]]], "-", RowBox[List["64", " ", SuperscriptBox["\[Nu]", "6"]]]]], RowBox[List["3072", " ", SuperscriptBox["z", "3"]]]], RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SqrtBox["2"], " ", "z"]], "-", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]]]], ")"]]]], "]"]]]], "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]

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> ker </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ∝ </mo> <mrow> <mfrac> <mrow> <msqrt> <mi> π </mi> </msqrt> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mo> - </mo> <mfrac> <mi> z </mi> <msqrt> <mn> 2 </mn> </msqrt> </mfrac> </mrow> </msup> </mrow> <msqrt> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 8 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msqrt> <mn> 2 </mn> </msqrt> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> ν </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mrow> <mn> 8 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 8 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msqrt> <mn> 2 </mn> </msqrt> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mrow> <mn> 16 </mn> <mo> ⁢ </mo> <msup> <mi> ν </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 40 </mn> <mo> ⁢ </mo> <msup> <mi> ν </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 9 </mn> </mrow> <mrow> <mn> 128 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 8 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> ⁢ </mo> <msqrt> <mn> 2 </mn> </msqrt> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 64 </mn> </mrow> <mo> ⁢ </mo> <msup> <mi> ν </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 560 </mn> <mo> ⁢ </mo> <msup> <mi> ν </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 1036 </mn> <mo> ⁢ </mo> <msup> <mi> ν </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 225 </mn> </mrow> <mrow> <mn> 3072 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 8 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msqrt> <mn> 2 </mn> </msqrt> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> - </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mo> … </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <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> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> KelvinKer </ci> <ci> ν </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <cos /> <apply> <times /> <cn type='rational'> 1 <sep /> 8 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> <apply> <times /> <pi /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> ν </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <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'> 8 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sin /> <apply> <times /> <cn type='rational'> 1 <sep /> 8 </cn> <apply> <plus /> <apply> <times /> <pi /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> ν </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 16 </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'> 40 </cn> <apply> <power /> <ci> ν </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 9 </cn> </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> <sin /> <apply> <times /> <cn type='rational'> 1 <sep /> 8 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -4 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> ν </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -64 </cn> <apply> <power /> <ci> ν </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 560 </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'> 1036 </cn> <apply> <power /> <ci> ν </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 225 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 3072 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <cos /> <apply> <times /> <cn type='rational'> 1 <sep /> 8 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <ci> ν </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <ci> … </ci> </apply> </apply> </apply> <apply> <ci> Rule </ci> <apply> <abs /> <ci> z </ci> </apply> <infinity /> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinKer", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["\[Pi]"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SqrtBox["2"], " ", "z"]], "+", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]]]], ")"]]]], "]"]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["4", " ", SuperscriptBox["\[Nu]", "2"]]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SqrtBox["2"], " ", "z"]], "+", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]]]], ")"]]]], "]"]]]], RowBox[List["8", " ", "z"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["9", "-", RowBox[List["40", " ", SuperscriptBox["\[Nu]", "2"]]], "+", RowBox[List["16", " ", SuperscriptBox["\[Nu]", "4"]]]]], ")"]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SqrtBox["2"], " ", "z"]], "-", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]]]], ")"]]]], "]"]]]], RowBox[List["128", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["225", "-", RowBox[List["1036", " ", SuperscriptBox["\[Nu]", "2"]]], "+", RowBox[List["560", " ", SuperscriptBox["\[Nu]", "4"]]], "-", RowBox[List["64", " ", SuperscriptBox["\[Nu]", "6"]]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SqrtBox["2"], " ", "z"]], "-", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]]]], ")"]]]], "]"]]]], RowBox[List["3072", " ", SuperscriptBox["z", "3"]]]], "+", "\[Ellipsis]"]], ")"]]]], SqrtBox[RowBox[List["2", " ", "z"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]

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

2007-05-02

KelvinKer[nu,z]