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BesselJ






Mathematica Notation

Traditional Notation









Bessel-Type Functions > BesselJ[nu,z] > Series representations > Generalized power series > Expansions at z==0 > For the function itself > General case





http://functions.wolfram.com/03.01.06.0037.01









  


  










Input Form





BesselJ[\[Nu], z] == Subscript[F, Infinity][z, \[Nu]] /; Subscript[F, n][z, \[Nu]] == Sum[((-1)^k (z/2)^(2 k + \[Nu]))/ (Gamma[k + \[Nu] + 1] k!), {k, 0, n}] == BesselJ[\[Nu], z] + (((-1)^n 2^(-\[Nu] - 2 n - 2) z^(\[Nu] + 2 n + 2))/ (Gamma[2 + n + \[Nu]] (n + 1)!)) HypergeometricPFQ[{1}, {2 + n, 2 + n + \[Nu]}, -(z^2/4)] && 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[SubscriptBox["F", "\[Infinity]"], "[", RowBox[List["z", ",", "\[Nu]"]], "]"]], "/;", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["F", "n"], "[", RowBox[List["z", ",", "\[Nu]"]], "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["(", FractionBox["z", "2"], ")"]], RowBox[List[RowBox[List["2", " ", "k"]], "+", "\[Nu]"]]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["k", "+", "\[Nu]", "+", "1"]], "]"]], " ", RowBox[List["k", "!"]]]]]]], "\[Equal]", RowBox[List[RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", RowBox[List["2", " ", "n"]], "-", "2"]]], " ", SuperscriptBox["z", RowBox[List["\[Nu]", "+", RowBox[List["2", " ", "n"]], "+", "2"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["2", "+", "n"]], ",", RowBox[List["2", "+", "n", "+", "\[Nu]"]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["2", "+", "n", "+", "\[Nu]"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], "!"]]]]]]]]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2007-05-02