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SphericalBesselJ






Mathematica Notation

Traditional Notation









Bessel-Type Functions > SphericalBesselJ[nu,z] > Differentiation > Symbolic differentiation > With respect to z





http://functions.wolfram.com/03.21.20.0020.01









  


  










Input Form





D[SphericalBesselJ[\[Nu], z], {z, n}] == 2^(1/2 - n) Pi n! Sum[((1/k!) 2^(2 k) HypergeometricPFQRegularized[{-(k/2), (1 - k)/2}, {1 - k + n, (1/4) (1 - 2 k), (1/4) (3 - 2 k)}, -(z^2/4)] SphericalBesselJ[k - n + \[Nu], z])/z^k, {k, 0, 2 n}] /; Element[n, Integers] && n >= 0










Standard Form





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MathML Form







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</ci> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "n_"]], "}"]]]]], RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], "-", "n"]]], " ", "\[Pi]", " ", RowBox[List["n", "!"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["2", " ", "n"]]], FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["2", " ", "k"]]], " ", SuperscriptBox["z", RowBox[List["-", "k"]]], " ", RowBox[List["HypergeometricPFQRegularized", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["k", "2"]]], ",", FractionBox[RowBox[List["1", "-", "k"]], "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", "k", "+", "n"]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["2", " ", "k"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "k"]]]], ")"]]]]]], "}"]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]]], "]"]], " ", RowBox[List["SphericalBesselJ", "[", RowBox[List[RowBox[List["k", "-", "n", "+", "\[Nu]"]], ",", "z"]], "]"]]]], RowBox[List["k", "!"]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2007-05-02





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