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BesselJ






Mathematica Notation

Traditional Notation









Bessel-Type Functions > BesselJ[nu,z] > Series representations > Generalized power series > Expansions at nu==+-n





http://functions.wolfram.com/03.01.06.0020.01









  


  










Input Form





BesselJ[\[Nu], z] \[Proportional] (-1)^n BesselJ[n, z] + ((-1)^(n - 1) Sum[((n - k - 1)!/k!) (z/2)^(2 k - n), {k, 0, n - 1}] + ((Pi (-1)^n)/2) BesselY[n, z] + ((n!/2) Sum[(1/((n - k) k!)) BesselJ[k, z] (z/2)^k, {k, 0, n - 1}])/ (-(z/2))^n + (1/n!) (-(z/2))^n Sum[(1/j) HypergeometricPFQ[{j}, {1 + j, 1 + n}, -(z^2/4)], {j, 1, n}]) (\[Nu] + n) + \[Ellipsis] /; (\[Nu] -> -n) && Element[n, Integers] && n > 0










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[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["BesselJ", "[", RowBox[List["n", ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "-", "1"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k", "-", "1"]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List[RowBox[List["2", " ", "k"]], "-", "n"]]]]], RowBox[List["k", "!"]]]]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]], " ", RowBox[List["BesselY", "[", RowBox[List["n", ",", "z"]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["n", "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["z", "2"]]], ")"]], RowBox[List["-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["BesselJ", "[", RowBox[List["k", ",", "z"]], "]"]], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], "k"]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], " ", RowBox[List["k", "!"]]]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["z", "2"]]], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], FractionBox[RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "j", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "+", "j"]], ",", RowBox[List["1", "+", "n"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]]], "]"]], "j"]]]]], RowBox[List["n", "!"]]]]], ")"]], " ", RowBox[List["(", RowBox[List["\[Nu]", "+", "n"]], ")"]]]], "+", "\[Ellipsis]"]], "/;", RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "\[Rule]", RowBox[List["-", "n"]]]], ")"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]










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





2007-05-02