Spherical Bessel function of the first kind: Representations through equivalent functions (formula 03.21.27.0003)

SphericalBesselJ

$Mathematica Notation$

Bessel-Type Functions

SphericalBesselJ[nu,z]

Representations through equivalent functions

With related functions

http://functions.wolfram.com/03.21.27.0003.01

Input Form

SphericalBesselJ[\[Nu], z] == Sec[Pi \[Nu]] SphericalBesselY[-1 - \[Nu], z] + Tan[Pi \[Nu]] SphericalBesselY[\[Nu], z]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["Sec", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["SphericalBesselY", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]], ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["Tan", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["SphericalBesselY", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msub> <mi> j </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <mrow> <mrow> <mi> sec </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <msub> <mi> y </mi> <mrow> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <msub> <mi> y </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> tan </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> SphericalBesselJ </ci> <ci> ν </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <sec /> <apply> <times /> <pi /> <ci> ν </ci> </apply> </apply> <apply> <ci> SphericalBesselY </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='integer'> -1 </cn> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <ci> SphericalBesselY </ci> <ci> ν </ci> <ci> z </ci> </apply> <apply> <tan /> <apply> <times /> <pi /> <ci> ν </ci> </apply> </apply> </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[RowBox[List["Sec", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["SphericalBesselY", "[", RowBox[List[RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]], ",", "z"]], "]"]]]], "+", RowBox[List[RowBox[List["Tan", "[", RowBox[List["\[Pi]", " ", "\[Nu]"]], "]"]], " ", RowBox[List["SphericalBesselY", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]]]]]]]]

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

2007-05-02