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BesselI






Mathematica Notation

Traditional Notation









Bessel-Type Functions > BesselI[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/03.02.06.0086.01









  


  










Input Form





BesselI[\[Nu], z] \[Proportional] Piecewise[{{(-(((-1)^(3/4) Sqrt[2])/(I^\[Nu] Sqrt[Pi] Sqrt[z]))) Cosh[z + (1/4) I Pi (1 + 2 \[Nu])], Arg[z] <= -(Pi/2)}, {(((-1)^(1/4) Sqrt[2] E^((I Pi \[Nu])/2))/(Sqrt[Pi] Sqrt[z])) Cosh[z - (1/4) I Pi (1 + 2 \[Nu])], Inequality[-(Pi/2), Less, Arg[z], LessEqual, Pi/2]}}, (((-1)^(3/4) Sqrt[2] E^((3 I Pi \[Nu])/2))/ (Sqrt[Pi] Sqrt[z])) Cosh[z + (1/4) I Pi (1 + 2 \[Nu])]] /; (Abs[z] -> Infinity)










Standard Form





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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[RowBox[List["\[Piecewise]", GridBox[List[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", SqrtBox["2"]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["z", "+", RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]]], "]"]]]], RowBox[List[SuperscriptBox["\[ImaginaryI]", "\[Nu]"], " ", SqrtBox["\[Pi]"], " ", SqrtBox["z"]]]]]], RowBox[List[RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", RowBox[List["-", FractionBox["\[Pi]", "2"]]]]]], List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", SqrtBox["2"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["z", "-", RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]]], "]"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", SqrtBox["z"]]]], RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "2"]]], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", FractionBox["\[Pi]", "2"]]]], List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["3", "/", "4"]]], " ", SqrtBox["2"], " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["3", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]], "2"]]]], ")"]], " ", RowBox[List["Cosh", "[", RowBox[List["z", "+", RowBox[List[FractionBox["1", "4"], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]]], "]"]]]], RowBox[List[SqrtBox["\[Pi]"], " ", SqrtBox["z"]]]], TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]]]], Rule[ColumnAlignments, List[Left]], Rule[ColumnSpacings, 1.2`], Rule[ColumnWidths, Automatic]]]], "/;", 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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