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SphericalBesselJ






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/03.21.20.0004.01









  


  










Input Form





Derivative[1, 0][SphericalBesselJ][-n, z] - (-(((-1)^n 2^(1 - n))/(z^n (n - 1)!))) (2 z Sum[(-4)^k z^(2 k) Binomial[-1 + n, 1 + 2 k] (-3 - 2 k + 2 n)! (CosIntegral[2 z] Sin[z] + (PolyGamma[3/2 + k] - PolyGamma[3/2 + k - n]) Sin[z] - Cos[z] SinIntegral[2 z]), {k, 0, -1 + Floor[n/2]}] + Sum[(-4)^k z^(2 k) Binomial[-1 + n, 2 k] (-2 - 2 k + 2 n)! (Cos[z] CosIntegral[2 z] + Cos[z] (PolyGamma[1/2 + k] - PolyGamma[3/2 + k - n]) + Sin[z] SinIntegral[2 z]), {k, 0, Floor[(n - 1)/2]}]) /; Element[n, Integers] && n > 0










Standard Form





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MathML Form







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</apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <cos /> <ci> z </ci> </apply> <apply> <ci> SinIntegral </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </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[RowBox[List[SuperscriptBox["SphericalBesselJ", TagBox[RowBox[List["(", RowBox[List["1", ",", "0"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List[RowBox[List["-", "n"]], ",", "z"]], "]"]], "-", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", SuperscriptBox["2", RowBox[List["1", "-", "n"]]], " ", SuperscriptBox["z", RowBox[List["-", "n"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "z", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "4"]], ")"]], "k"], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ",", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "-", RowBox[List["2", " ", "k"]], "+", RowBox[List["2", " ", "n"]]]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["CosIntegral", "[", RowBox[List["2", " ", "z"]], "]"]], " ", RowBox[List["Sin", "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["3", "2"], "+", "k"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["3", "2"], "+", "k", "-", "n"]], "]"]]]], ")"]], " ", RowBox[List["Sin", "[", "z", "]"]]]], "-", RowBox[List[RowBox[List["Cos", "[", "z", "]"]], " ", RowBox[List["SinIntegral", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], ")"]]]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "4"]], ")"]], "k"], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "n"]], ",", RowBox[List["2", " ", "k"]]]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["2", " ", "k"]], "+", RowBox[List["2", " ", "n"]]]], ")"]], "!"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Cos", "[", "z", "]"]], " ", RowBox[List["CosIntegral", "[", RowBox[List["2", " ", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["Cos", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["1", "2"], "+", "k"]], "]"]], "-", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox["3", "2"], "+", "k", "-", "n"]], "]"]]]], ")"]]]], "+", RowBox[List[RowBox[List["Sin", "[", "z", "]"]], " ", RowBox[List["SinIntegral", "[", RowBox[List["2", " ", "z"]], "]"]]]]]], ")"]]]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]










Contributed by





Brychkov Yu.A. (2005)










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





2007-05-02





© 1998-2014 Wolfram Research, Inc.