Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











SphericalBesselJ






Mathematica Notation

Traditional Notation









Bessel-Type Functions > SphericalBesselJ[nu,z] > Transformations > Addition formulas





http://functions.wolfram.com/03.21.16.0006.01









  


  










Input Form





SphericalBesselJ[\[Nu], Subscript[z, 1] + Subscript[z, 2]] == Sqrt[2/Pi] ((Sqrt[Subscript[z, 1]] Sqrt[Subscript[z, 2]])/ Sqrt[Subscript[z, 1] + Subscript[z, 2]]) Sum[SphericalBesselJ[-(1/2) + k, Subscript[z, 2]] SphericalBesselJ[-k + \[Nu], Subscript[z, 1]], {k, -Infinity, Infinity}] /; Abs[Subscript[z, 2]/Subscript[z, 1]] < 1 || Element[\[Nu] + 1/2, Integers]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]", ",", RowBox[List[SubscriptBox["z", "1"], "+", SubscriptBox["z", "2"]]]]], "]"]], " ", "\[Equal]", RowBox[List[SqrtBox[FractionBox["2", "\[Pi]"]], FractionBox[RowBox[List[" ", RowBox[List[SqrtBox[SubscriptBox["z", "1"]], " ", SqrtBox[SubscriptBox["z", "2"]]]]]], SqrtBox[RowBox[List[SubscriptBox["z", "1"], "+", SubscriptBox["z", "2"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List[RowBox[List["SphericalBesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "k"]], ",", SubscriptBox["z", "2"]]], "]"]], " ", RowBox[List["SphericalBesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", "k"]], "+", "\[Nu]"]], ",", SubscriptBox["z", "1"]]], "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", FractionBox[SubscriptBox["z", "2"], SubscriptBox["z", "1"]], "]"]], "<", "1"]], "\[Or]", RowBox[List[RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], "\[Element]", "Integers"]]]]]]]]










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> <msub> <mi> j </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <msqrt> <mfrac> <mn> 2 </mn> <mi> &#960; </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <mfrac> <mrow> <mtext> </mtext> <mrow> <msqrt> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </msqrt> <mo> &#8290; </mo> <msqrt> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </msqrt> </mrow> </mrow> <msqrt> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> + </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mrow> <msub> <mi> j </mi> <mrow> <mi> k </mi> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msub> <mo> ( </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> j </mi> <mrow> <mi> &#957; </mi> <mo> - </mo> <mi> k </mi> </mrow> </msub> <mo> ( </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mfrac> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mfrac> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mn> 1 </mn> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mi> &#957; </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[List[], Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> SphericalBesselJ </ci> <ci> &#957; </ci> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <infinity /> </apply> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> SphericalBesselJ </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> SphericalBesselJ </ci> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <or /> <apply> <lt /> <apply> <abs /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <in /> <apply> <plus /> <ci> &#957; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <integers /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["SphericalBesselJ", "[", RowBox[List["\[Nu]_", ",", RowBox[List[SubscriptBox["z_", "1"], "+", SubscriptBox["z_", "2"]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SqrtBox[FractionBox["2", "\[Pi]"]], " ", RowBox[List["(", RowBox[List[SqrtBox[SubscriptBox["zz", "1"]], " ", SqrtBox[SubscriptBox["zz", "2"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["-", "\[Infinity]"]]]], "\[Infinity]"], RowBox[List[RowBox[List["SphericalBesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "k"]], ",", SubscriptBox["zz", "2"]]], "]"]], " ", RowBox[List["SphericalBesselJ", "[", RowBox[List[RowBox[List[RowBox[List["-", "k"]], "+", "\[Nu]"]], ",", SubscriptBox["zz", "1"]]], "]"]]]]]]]], SqrtBox[RowBox[List[SubscriptBox["zz", "1"], "+", SubscriptBox["zz", "2"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", FractionBox[SubscriptBox["zz", "2"], SubscriptBox["zz", "1"]], "]"]], "<", "1"]], "||", RowBox[List[RowBox[List["\[Nu]", "+", FractionBox["1", "2"]]], "\[Element]", "Integers"]]]]]]]]]]










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





2007-05-02





© 1998- Wolfram Research, Inc.