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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 exponential form > Using exponential function with branch cut-free arguments





http://functions.wolfram.com/03.02.06.0076.01









  


  










Input Form





BesselI[\[Nu], z] \[Proportional] (((-1)^(1/4) z^\[Nu] I^\[Nu])/((I z)^\[Nu] Sqrt[2 Pi I z])) (E^z (1 + (1 - 4 \[Nu]^2)/(8 z) + (9 - 40 \[Nu]^2 + 16 \[Nu]^4)/ (128 z^2) + \[Ellipsis]) + ((I (Sqrt[z^2]/z) Cos[Pi \[Nu]] - Sin[Pi \[Nu]]) (1 + (-1 + 4 \[Nu]^2)/(8 z) + (9 - 40 \[Nu]^2 + 16 \[Nu]^4)/(128 z^2) + \[Ellipsis]))/E^z) /; (Abs[z] -> Infinity)










Standard Form





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MathML Form







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</mo> <msup> <mi> &#957; </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 9 </mn> <mo> - </mo> <mrow> <mn> 40 </mn> <mo> &#8290; </mo> <msup> <mi> &#957; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 16 </mn> <mo> &#8290; </mo> <msup> <mi> &#957; </mi> <mn> 4 </mn> </msup> </mrow> </mrow> <mrow> <mn> 128 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> </mrow> <mi> z </mi> </mfrac> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> sin </mi> <mo> &#8289; 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</mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> BesselI </ci> <ci> &#957; </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <ci> &#957; </ci> </apply> <apply> <power /> <imaginaryi /> <ci> &#957; </ci> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <imaginaryi /> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <ci> z </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <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> &#957; </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> <times /> <apply> <plus /> <cn type='integer'> 9 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 40 </cn> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 4 </cn> </apply> </apply> </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> <ci> &#8230; </ci> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <cos /> <apply> <times /> <pi /> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sin /> <apply> <times /> <pi /> <ci> &#957; </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 8 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 9 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 40 </cn> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 4 </cn> </apply> </apply> </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> <ci> &#8230; </ci> </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["BesselI", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "z"]], ")"]], RowBox[List["-", "\[Nu]"]]], " ", SuperscriptBox["z", "\[Nu]"], " ", SuperscriptBox["\[ImaginaryI]", "\[Nu]"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", "z"], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["1", "-", RowBox[List["4", " ", SuperscriptBox["\[Nu]", "2"]]]]], RowBox[List["8", " ", "z"]]], "+", FractionBox[RowBox[List["9", "-", RowBox[List["40", " ", SuperscriptBox["\[Nu]", "2"]]], "+", RowBox[List["16", " ", SuperscriptBox["\[Nu]", "4"]]]]], RowBox[List["128", " ", SuperscriptBox["z", "2"]]]], "+", "\[Ellipsis]"]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "z"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox[SuperscriptBox["z", "2"]], " ", RowBox[List["Cos", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], "z"], "-", RowBox[List["Sin", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["4", " ", SuperscriptBox["\[Nu]", "2"]]]]], RowBox[List["8", " ", "z"]]], "+", FractionBox[RowBox[List["9", "-", RowBox[List["40", " ", SuperscriptBox["\[Nu]", "2"]]], "+", RowBox[List["16", " ", SuperscriptBox["\[Nu]", "4"]]]]], RowBox[List["128", " ", SuperscriptBox["z", "2"]]]], "+", "\[Ellipsis]"]], ")"]]]]]], ")"]]]], SqrtBox[RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "z"]]]], "/;", 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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