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SphericalBesselJ






Mathematica Notation

Traditional Notation









Bessel-Type Functions > SphericalBesselJ[nu,z] > Differential equations > Ordinary linear differential equations and wronskians > For the direct function itself





http://functions.wolfram.com/03.21.13.0013.01









  


  










Input Form





z^2 Derivative[2][w][z] + z (1 + r - 2 s) Derivative[1][w][z] + ((-r) s + s^2 + r^2 (a^2 z^(2 r) - \[Nu] (1 + \[Nu]))) w[z] == 0 /; w[z] == Subscript[c, 1] z^s SphericalBesselJ[\[Nu], a z^r] + Subscript[c, 2] z^s SphericalBesselY[\[Nu], a z^r]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["1", "+", "r", "-", RowBox[List["2", " ", "s"]]]], ")"]], " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "r"]], " ", "s"]], "+", SuperscriptBox["s", "2"], "+", RowBox[List[SuperscriptBox["r", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["a", "2"], " ", SuperscriptBox["z", RowBox[List["2", " ", "r"]]]]], "-", RowBox[List["\[Nu]", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["w", "[", "z", "]"]]]]]], "\[Equal]", "0"]], "/;", " ", RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], SuperscriptBox["z", "s"], RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], SuperscriptBox["z", "s"], RowBox[List["SphericalBesselY", "[", RowBox[List["\[Nu]", ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]]]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <mrow> <msup> <mi> w </mi> <mi> &#8242;&#8242; </mi> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> r </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> s </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> w </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> r </mi> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <mi> &#957; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> r </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> s </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mi> r </mi> <mo> &#8290; </mo> <mi> s </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> w </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#63449; </mo> <mn> 0 </mn> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> w </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mrow> <msub> <mi> c </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> s </mi> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> j </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msub> <mi> c </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> s </mi> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> y </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mi> r </mi> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <partialdiff /> <bvar> <ci> z </ci> <degree> <cn type='integer'> 2 </cn> </degree> </bvar> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <ci> z </ci> <apply> <plus /> <ci> r </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> s </ci> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <partialdiff /> <bvar> <ci> z </ci> </bvar> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> r </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#957; </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> r </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <ci> s </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> r </ci> <ci> s </ci> </apply> </apply> </apply> <apply> <ci> w </ci> <ci> z </ci> </apply> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <eq /> <apply> <ci> w </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> z </ci> <ci> s </ci> </apply> <apply> <ci> SphericalBesselJ </ci> <ci> &#957; </ci> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <ci> s </ci> </apply> <apply> <ci> SphericalBesselY </ci> <ci> &#957; </ci> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <ci> r </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List[SuperscriptBox["z_", "2"], " ", RowBox[List[SuperscriptBox["w", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "+", RowBox[List["z_", " ", RowBox[List["(", RowBox[List["1", "+", "r_", "-", RowBox[List["2", " ", "s_"]]]], ")"]], " ", RowBox[List[SuperscriptBox["w", "\[Prime]", Rule[MultilineFunction, None]], "[", "z_", "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "r_"]], " ", "s_"]], "+", SuperscriptBox["s_", "2"], "+", RowBox[List[SuperscriptBox["r_", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["a_", "2"], " ", SuperscriptBox["z_", RowBox[List["2", " ", "r_"]]]]], "-", RowBox[List["\[Nu]_", " ", RowBox[List["(", RowBox[List["1", "+", "\[Nu]_"]], ")"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["w", "[", "z_", "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["0", "/;", RowBox[List[RowBox[List["w", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[SubscriptBox["c", "1"], " ", SuperscriptBox["z", "s"], " ", RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]]]], "+", RowBox[List[SubscriptBox["c", "2"], " ", SuperscriptBox["z", "s"], " ", RowBox[List["SphericalBesselY", "[", RowBox[List["\[Nu]", ",", RowBox[List["a", " ", SuperscriptBox["z", "r"]]]]], "]"]]]]]]]]]]]]]]










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





2007-05-02





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