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variants of this functions
KelvinBer






Mathematica Notation

Traditional Notation









Bessel-Type Functions > KelvinBer[z] > Series representations > Asymptotic series expansions > Expansions inside Stokes sectors > Expansions containing z->infinity > In trigonometric form ||| In trigonometric form





http://functions.wolfram.com/03.14.06.0022.01









  


  










Input Form





KelvinBer[z] \[Proportional] (1/(Sqrt[2 Pi] Sqrt[z])) (E^(z/Sqrt[2]) Cos[(1/8) (Pi - 4 Sqrt[2] z)] + (I Sin[(1/8) (3 Pi - 4 Sqrt[2] z)])/E^(z/Sqrt[2]) + (1/(8 z)) (E^(z/Sqrt[2]) Sin[(1/8) (Pi + 4 Sqrt[2] z)] + (I Sin[(1/8) (-Pi + 4 Sqrt[2] z)])/E^(z/Sqrt[2])) + (9/(128 z^2)) (E^(z/Sqrt[2]) Sin[(1/8) (-Pi + 4 Sqrt[2] z)] - (I Cos[(1/8) (-3 Pi + 4 Sqrt[2] z)])/E^(z/Sqrt[2])) - (75/(1024 z^3)) (E^(z/Sqrt[2]) Cos[(1/8) (-Pi - 4 Sqrt[2] z)] - (I Cos[(1/8) (Pi - 4 Sqrt[2] z)])/E^(z/Sqrt[2])) + \[Ellipsis]) /; Inequality[-(Pi/2), Less, Arg[z], LessEqual, Pi] && (Abs[z] -> Infinity)










Standard Form





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MathML Form







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type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <cos /> <apply> <times /> <cn type='rational'> 1 <sep /> 8 </cn> <apply> <plus /> <pi /> <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> <times /> <imaginaryi /> <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> <sin /> <apply> <times /> <cn type='rational'> 1 <sep /> 8 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <pi /> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn 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<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> <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> <pi /> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 9 </cn> <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> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> z </ci> 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<ci> &#8230; </ci> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Inequality </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <lt /> <apply> <arg /> <ci> z </ci> </apply> <leq /> <pi /> </apply> <apply> <ci> Rule </ci> <apply> <abs /> <ci> z </ci> </apply> <infinity /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["KelvinBer", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["3", " ", "\[Pi]"]], "-", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "]"]]]], "+", FractionBox[RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List["\[Pi]", "+", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "]"]]]], "+", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[Pi]"]], "+", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "]"]]]]]], RowBox[List["8", " ", "z"]]], "+", FractionBox[RowBox[List["9", " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[Pi]"]], "+", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "3"]], " ", "\[Pi]"]], "+", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "]"]]]]]], ")"]]]], RowBox[List["128", " ", SuperscriptBox["z", "2"]]]], "-", FractionBox[RowBox[List["75", " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", FractionBox["z", SqrtBox["2"]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[Pi]"]], "-", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "]"]]]], "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox["z", SqrtBox["2"]]]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "8"], " ", RowBox[List["(", RowBox[List["\[Pi]", "-", RowBox[List["4", " ", SqrtBox["2"], " ", "z"]]]], ")"]]]], "]"]]]]]], ")"]]]], RowBox[List["1024", " ", SuperscriptBox["z", "3"]]]], "+", "\[Ellipsis]"]], RowBox[List[SqrtBox[RowBox[List["2", " ", "\[Pi]"]]], " ", SqrtBox["z"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "2"]]], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "\[Pi]"]], "&&", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]]]










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





2007-05-02