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

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











BesselJ






Mathematica Notation

Traditional Notation









Bessel-Type Functions > BesselJ[nu,z] > Series representations > Generalized power series > Expansions at generic point z==z0 > For the function itself





http://functions.wolfram.com/03.01.06.0022.01









  


  










Input Form





BesselJ[\[Nu], z] \[Proportional] (1/Subscript[z, 0])^(\[Nu] Floor[Arg[z - Subscript[z, 0]]/(2 Pi)]) Subscript[z, 0]^(\[Nu] Floor[Arg[z - Subscript[z, 0]]/(2 Pi)]) (BesselJ[\[Nu], Subscript[z, 0]] + (-BesselJ[1 + \[Nu], Subscript[z, 0]] + (\[Nu]/Subscript[z, 0]) BesselJ[\[Nu], Subscript[z, 0]]) (z - Subscript[z, 0]) + ((((-1 + \[Nu]) \[Nu] - Subscript[z, 0]^2) BesselJ[\[Nu], Subscript[z, 0]] + BesselJ[1 + \[Nu], Subscript[z, 0]] Subscript[z, 0])/(2 Subscript[z, 0]^2)) (z - Subscript[z, 0])^2 + O[(z - Subscript[z, 0])^3])










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["1", SubscriptBox["z", "0"]], ")"]], RowBox[List["\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SubsuperscriptBox["z", "0", RowBox[List["\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["z", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", SubscriptBox["z", "0"]]], "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", SubscriptBox["z", "0"]]], "]"]]]], "+", RowBox[List[FractionBox["\[Nu]", SubscriptBox["z", "0"]], RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", SubscriptBox["z", "0"]]], "]"]]]]]], ")"]], RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ")"]], " ", "\[Nu]"]], " ", "-", SubsuperscriptBox["z", "0", "2"]]], ")"]], RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", SubscriptBox["z", "0"]]], "]"]]]], "+", RowBox[List[RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", SubscriptBox["z", "0"]]], "]"]], " ", SubscriptBox["z", "0"]]]]], RowBox[List["2", " ", SubsuperscriptBox["z", "0", "2"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "2"]]], "+", RowBox[List["O", "[", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], "3"], "]"]]]], ")"]]]]]]]]










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> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mfrac> <mn> 1 </mn> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> &#957; </mi> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 0 </mn> <mrow> <mi> &#957; </mi> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> </msubsup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> J </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mi> &#957; </mi> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <mi> J </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msub> <mi> J </mi> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mrow> <mrow> <msub> <mi> J </mi> <mrow> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mrow> <msub> <mi> J </mi> <mi> &#957; </mi> </msub> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> &#957; </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> - </mo> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msubsup> <mi> z </mi> <mn> 0 </mn> <mn> 2 </mn> </msubsup> </mrow> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Proportional </ci> <apply> <ci> BesselJ </ci> <ci> &#957; </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> &#957; </ci> <apply> <floor /> <apply> <times /> <apply> <arg /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </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> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <apply> <times /> <ci> &#957; </ci> <apply> <floor /> <apply> <times /> <apply> <arg /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </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> BesselJ </ci> <ci> &#957; </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> BesselJ </ci> <ci> &#957; </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> BesselJ </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <ci> BesselJ </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <times /> <apply> <ci> BesselJ </ci> <ci> &#957; </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> -1 </cn> </apply> <ci> &#957; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <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> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["1", SubscriptBox["zz", "0"]], ")"]], RowBox[List["\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", SubsuperscriptBox["zz", "0", RowBox[List["\[Nu]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", SubscriptBox["zz", "0"]]], "]"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", SubscriptBox["zz", "0"]]], "]"]]]], "+", FractionBox[RowBox[List["\[Nu]", " ", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", SubscriptBox["zz", "0"]]], "]"]]]], SubscriptBox["zz", "0"]]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ")"]], " ", "\[Nu]"]], "-", SubsuperscriptBox["zz", "0", "2"]]], ")"]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", SubscriptBox["zz", "0"]]], "]"]]]], "+", RowBox[List[RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["1", "+", "\[Nu]"]], ",", SubscriptBox["zz", "0"]]], "]"]], " ", SubscriptBox["zz", "0"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "2"]]], RowBox[List["2", " ", SubsuperscriptBox["zz", "0", "2"]]]], "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], "]"]], "3"]]], ")"]]]]]]]]










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





2007-05-02





© 1998-2014 Wolfram Research, Inc.