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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/03.02.06.0079.01









  


  










Input Form





BesselI[\[Nu], z] \[Proportional] (1/Sqrt[4 Pi]) z^\[Nu] (-z^2)^((1/4) (-1 - 2 \[Nu])) ((HypergeometricPFQ[{1/2 + \[Nu], 1/2 - \[Nu]}, {}, -(1/(2 z))] ((1 + z/Sqrt[-z^2]) Cos[(Pi \[Nu])/2] - (1 - z/Sqrt[-z^2]) Sin[(Pi \[Nu])/2]))/E^z + E^z HypergeometricPFQ[ {1/2 + \[Nu], 1/2 - \[Nu]}, {}, 1/(2 z)] ((1 - z/Sqrt[-z^2]) Cos[(Pi \[Nu])/2] - (1 + z/Sqrt[-z^2]) Sin[(Pi \[Nu])/2])) /; (Abs[z] -> Infinity)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[FractionBox["1", SqrtBox[RowBox[List["4", " ", "\[Pi]"]]]], SuperscriptBox["z", "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "z"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["-", FractionBox["1", RowBox[List["2", " ", "z"]]]]]]], "]"]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", FractionBox["z", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]]], ")"]], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["1", "-", FractionBox["z", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]]], ")"]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", "z"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", FractionBox["1", RowBox[List["2", " ", "z"]]]]], "]"]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", FractionBox["z", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]]], ")"]], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["1", "+", FractionBox["z", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]]], ")"]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]]]]]], " ", ")"]]]]]], " ", ")"]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msub> <mi> I </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mfrac> <mrow> <msup> <mi> z </mi> <mi> &#957; </mi> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> <msqrt> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msup> <mi> &#8519; </mi> <mi> z </mi> </msup> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mi> z </mi> <msqrt> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; 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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[SuperscriptBox["z", "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "z"]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["-", FractionBox["1", RowBox[List["2", " ", "z"]]]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", FractionBox["z", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]]], ")"]], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["1", "-", FractionBox["z", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]]], ")"]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]]]]]], ")"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", "z"], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", FractionBox["1", RowBox[List["2", " ", "z"]]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "-", FractionBox["z", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]]], ")"]], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["1", "+", FractionBox["z", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]]], ")"]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]]]]]], ")"]]]]]], ")"]]]], SqrtBox[RowBox[List["4", " ", "\[Pi]"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]










Contributed by





Pavlyk 0. (2006)










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





2007-05-02





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