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BesselJ






Mathematica Notation

Traditional Notation









Bessel-Type Functions > BesselJ[nu,z] > Series representations > Asymptotic series expansions > Expansions for any z in exponential form > Using exponential function with branch cut-containing arguments





http://functions.wolfram.com/03.01.06.0007.01









  


  










Input Form





BesselJ[\[Nu], z] \[Proportional] ((1/Sqrt[2 Pi]) z^\[Nu] (HypergeometricPFQ[{1/2 + \[Nu], 1/2 - \[Nu]}, {}, -(I/(2 Sqrt[z^2]))]/E^(I (((2 \[Nu] + 1)/4) Pi - Sqrt[z^2])) + E^(I (((2 \[Nu] + 1)/4) Pi - Sqrt[z^2])) HypergeometricPFQ[ {1/2 + \[Nu], 1/2 - \[Nu]}, {}, I/(2 Sqrt[z^2])]))/ (z^2)^((2 \[Nu] + 1)/4) /; (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["BesselJ", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]], " ", "\[Pi]"]], "-", SqrtBox[SuperscriptBox["z", "2"]]]], ")"]]]]], " ", 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["\[ImaginaryI]", RowBox[List["2", " ", SqrtBox[SuperscriptBox["z", "2"]]]]]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[Nu]"]], "+", "1"]], ")"]], " ", "\[Pi]"]], "-", SqrtBox[SuperscriptBox["z", "2"]]]], ")"]]]]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List[FractionBox["1", "2"], "-", "\[Nu]"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", FractionBox["\[ImaginaryI]", RowBox[List["2", " ", SqrtBox[SuperscriptBox["z", "2"]]]]]]], "]"]]]]]], ")"]]]], SqrtBox[RowBox[List["2", " ", "\[Pi]"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]










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





2001-10-29