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variants of this functions
KelvinKer






Mathematica Notation

Traditional 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/03.16.06.0021.01









  


  










Input Form





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










Standard Form





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MathML Form







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</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> <ci> HypergeometricPFQ </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <ci> O </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </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[RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]], "+", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["SeriesData", "[", RowBox[List["z", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", "4"]], "]"]]]], ")"]]]], "-", 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["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]], "+", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["SeriesData", "[", RowBox[List["z", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", "4"]], "]"]]]], ")"]]]], RowBox[List["128", " ", SuperscriptBox["z", "2"]]]], "-", 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["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]], "-", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["SeriesData", "[", RowBox[List["z", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", "4"]], "]"]]]], ")"]]]], RowBox[List["8", " ", "z"]]], "+", 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["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["4", " ", "\[Nu]"]]]], ")"]]]], "-", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["SeriesData", "[", RowBox[List["z", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", "4"]], "]"]]]], ")"]]]], RowBox[List["3072", " ", SuperscriptBox["z", "3"]]]]]], ")"]]]], 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





© 1998- Wolfram Research, Inc.