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BesselI






Mathematica Notation

Traditional Notation









Bessel-Type Functions > BesselI[nu,z] > Series representations > Asymptotic series expansions > Expansions inside Stokes sectors > Expansions containing z->infinity > In hyperbolic form ||| In hyperbolic form





http://functions.wolfram.com/03.02.06.0063.01









  


  










Input Form





BesselI[\[Nu], z] \[Proportional] ((Sqrt[2]/Sqrt[Pi z]) (Cosh[z + (Pi I (2 \[Nu] + 1))/4] (1 + (9 - 40 \[Nu]^2 + 16 \[Nu]^4)/(128 z^2) + (11025 - 51664 \[Nu]^2 + 31584 \[Nu]^4 - 5376 \[Nu]^6 + 256 \[Nu]^8)/ (98304 z^4) + \[Ellipsis]) + ((1 - 4 \[Nu]^2)/(8 z)) Sinh[z + (Pi I (2 \[Nu] + 1))/4] (1 + (225 - 136 \[Nu]^2 + 16 \[Nu]^4)/ (384 z^2) + (893025 - 656784 \[Nu]^2 + 137824 \[Nu]^4 - 10496 \[Nu]^6 + 256 \[Nu]^8)/(491520 z^4) + \[Ellipsis])))/ E^((Pi I (2 \[Nu] + 1))/4) /; -Pi < Arg[z] < Pi/2 && (Abs[z] -> Infinity)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[FractionBox[SqrtBox["2"], SqrtBox[RowBox[List["\[Pi]", " ", "z"]]]], SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", RowBox[List["(", RowBox[List[RowBox[List["2", "\[Nu]"]], "+", "1"]], ")"]]]], "4"]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cosh", "[", RowBox[List["z", "+", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]]]], "4"]]], "]"]], RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["9", "-", RowBox[List["40", " ", SuperscriptBox["\[Nu]", "2"]]], "+", RowBox[List["16", " ", SuperscriptBox["\[Nu]", "4"]]]]], 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["1", "-", RowBox[List["4", " ", SuperscriptBox["\[Nu]", "2"], " "]]]], RowBox[List["8", " ", "z"]]], RowBox[List["Sinh", "[", RowBox[List["z", "+", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]]]], "4"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["225", "-", RowBox[List["136", " ", SuperscriptBox["\[Nu]", "2"]]], "+", RowBox[List["16", " ", SuperscriptBox["\[Nu]", "4"]]]]], 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["-", "\[Pi]"]], "<", RowBox[List["Arg", "[", "z", "]"]], "<", FractionBox["\[Pi]", "2"]]], "\[And]", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]










MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BesselI", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SqrtBox["2"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]]]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cosh", "[", RowBox[List["z", "+", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]]]]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["9", "-", RowBox[List["40", " ", SuperscriptBox["\[Nu]", "2"]]], "+", RowBox[List["16", " ", SuperscriptBox["\[Nu]", "4"]]]]], 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]"]], ")"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", RowBox[List["4", " ", SuperscriptBox["\[Nu]", "2"]]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["z", "+", RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]]]]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["225", "-", RowBox[List["136", " ", SuperscriptBox["\[Nu]", "2"]]], "+", RowBox[List["16", " ", SuperscriptBox["\[Nu]", "4"]]]]], 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["8", " ", "z"]]]]], ")"]]]], SqrtBox[RowBox[List["\[Pi]", " ", "z"]]]], "/;", RowBox[List[RowBox[List[RowBox[List["-", "\[Pi]"]], "<", RowBox[List["Arg", "[", "z", "]"]], "<", FractionBox["\[Pi]", "2"]]], "&&", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]]]










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





2007-05-02





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