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SphericalBesselJ




Mathematica Notation

Traditional Notation









Bessel-Type Functions > SphericalBesselJ[nu,z] > Integration > Indefinite integration > Involving functions of the direct function and elementary functions > Involving elementary functions of the direct function and elementary functions > Involving powers of the direct function and a power function > Linear arguments





http://functions.wolfram.com/03.21.21.0050.01









  


  








Input Form





Integrate[SphericalBesselJ[\[Nu], a z]^2/z, z] == (1/(3 + 8 \[Nu] + 4 \[Nu]^2)) (2 a^2 z^2 SphericalBesselJ[-(1/2) + \[Nu], a z]^2 - 2 a z (1 + 2 \[Nu]) SphericalBesselJ[-(1/2) + \[Nu], a z] SphericalBesselJ[1/2 + \[Nu], a z] + (-1 + 2 a^2 z^2 - 2 \[Nu]) SphericalBesselJ[1/2 + \[Nu], a z]^2)








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", RowBox[List["a", " ", "z"]]]], "]"]], "2"], "z"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["3", "+", RowBox[List["8", " ", "\[Nu]"]], "+", RowBox[List["4", " ", SuperscriptBox["\[Nu]", "2"]]]]]], RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"], " ", SuperscriptBox[RowBox[List["SphericalBesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "\[Nu]"]], ",", RowBox[List["a", " ", "z"]]]], "]"]], "2"]]], "-", RowBox[List["2", " ", "a", " ", "z", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["SphericalBesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "\[Nu]"]], ",", RowBox[List["a", " ", "z"]]]], "]"]], " ", RowBox[List["SphericalBesselJ", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["a", " ", "z"]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["SphericalBesselJ", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["a", " ", "z"]]]], "]"]], "2"]]]]], ")"]]]]]]]]








MathML Form







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</mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> j </mi> <mrow> <mi> &#957; </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> j </mi> <mrow> <mi> &#957; </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mrow> <msub> <mi> j </mi> <mrow> <mi> &#957; 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</ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <ci> SphericalBesselJ </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <ci> SphericalBesselJ </ci> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <ci> SphericalBesselJ </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> SphericalBesselJ </ci> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <ci> a </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 8 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]_", ",", RowBox[List["a_", " ", "z_"]]]], "]"]], "2"], "z_"], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"], " ", SuperscriptBox[RowBox[List["SphericalBesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "\[Nu]"]], ",", RowBox[List["a", " ", "z"]]]], "]"]], "2"]]], "-", RowBox[List["2", " ", "a", " ", "z", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["SphericalBesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "\[Nu]"]], ",", RowBox[List["a", " ", "z"]]]], "]"]], " ", RowBox[List["SphericalBesselJ", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["a", " ", "z"]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", SuperscriptBox["a", "2"], " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["SphericalBesselJ", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "\[Nu]"]], ",", RowBox[List["a", " ", "z"]]]], "]"]], "2"]]]]], RowBox[List["3", "+", RowBox[List["8", " ", "\[Nu]"]], "+", RowBox[List["4", " ", SuperscriptBox["\[Nu]", "2"]]]]]]]]]]


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





2007-05-02