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SphericalBesselJ






Mathematica Notation

Traditional Notation









Bessel-Type Functions > SphericalBesselJ[nu,z] > Series representations > Generalized power series > Expansions on branch cuts > For the function itself





http://functions.wolfram.com/03.21.06.0010.01









  


  










Input Form





SphericalBesselJ[\[Nu], z] \[Proportional] Exp[2 \[Nu] Pi I Floor[Arg[z - x]/(2 Pi)]] (SphericalBesselJ[\[Nu], x] + (SphericalBesselJ[-1 + \[Nu], x] - ((1 + \[Nu])/x) SphericalBesselJ[\[Nu], x]) (z - x) + ((-2 x SphericalBesselJ[-1 + \[Nu], x] + (2 - x^2 + 3 \[Nu] + \[Nu]^2) SphericalBesselJ[\[Nu], x])/(2 x^2)) (z - x)^2 + \[Ellipsis]) /; (z -> x) && Element[x, Reals] && x < 0










Standard Form





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MathML Form







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</mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> x </mi> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <mi> j </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mi> x </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> &#957; </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> j </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> x </mi> <mo> &#8290; </mo> <mrow> <msub> <mi> j </mi> <mrow> <mi> &#957; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; 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</ci> <pi /> <imaginaryi /> <apply> <floor /> <apply> <times /> <apply> <arg /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <ci> SphericalBesselJ </ci> <ci> &#957; </ci> <ci> x </ci> </apply> <apply> <times /> <apply> <plus /> <apply> <ci> SphericalBesselJ </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> -1 </cn> </apply> <ci> x </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> SphericalBesselJ </ci> <ci> &#957; </ci> <ci> x </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> SphericalBesselJ </ci> <ci> &#957; </ci> <ci> x </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> x </ci> <apply> <ci> SphericalBesselJ </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> -1 </cn> </apply> <ci> x </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> z </ci> <ci> x </ci> </apply> <apply> <in /> <ci> x </ci> <reals /> </apply> <apply> <lt /> <ci> x </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[Nu]", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["SphericalBesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ",", "x"]], "]"]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], " ", RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]]]], "x"]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "x", " ", RowBox[List["SphericalBesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ",", "x"]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["2", "-", SuperscriptBox["x", "2"], "+", RowBox[List["3", " ", "\[Nu]"]], "+", SuperscriptBox["\[Nu]", "2"]]], ")"]], " ", RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", "x"]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "2"]]], RowBox[List["2", " ", SuperscriptBox["x", "2"]]]], "+", "\[Ellipsis]"]], ")"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", "x"]], ")"]], "&&", RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["x", "<", "0"]]]]]]]]]]










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





2007-05-02