Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail Comments

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] > Integral transforms > Fourier exp transforms





http://functions.wolfram.com/03.01.22.0006.01









  


  










Input Form





FourierTransform[Subscript[j, n][t], t, z] == Sqrt[Pi/2] I^n UnitStep[1 - z^2] LegendreP[n, z] /; Element[n, Integers] && n >= 0 && Subscript[j, n][z] == Sqrt[Pi/(2 z)] BesselJ[n + 1/2, z] && Element[z, Reals]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["FourierTransform", "[", RowBox[List[RowBox[List[SubscriptBox["j", "n"], "[", "t", "]"]], ",", "t", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SqrtBox[FractionBox["\[Pi]", "2"]], SuperscriptBox["\[ImaginaryI]", "n"], RowBox[List["UnitStep", "[", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], "]"]], RowBox[List["LegendreP", "[", RowBox[List["n", ",", "z"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List[RowBox[List[SubscriptBox["j", "n"], "[", "z", "]"]], "\[Equal]", RowBox[List[SqrtBox[FractionBox["\[Pi]", RowBox[List["2", " ", "z"]]]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["n", "+", FractionBox["1", "2"]]], ",", "z"]], "]"]]]]]], "\[And]", RowBox[List["z", "\[Element]", "Reals"]]]]]]]]










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> <msub> <mi> &#8497; </mi> <mi> t </mi> </msub> <mo> [ </mo> <mrow> <msub> <mi> j </mi> <mi> n </mi> </msub> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> ] </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <msqrt> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </msqrt> <mo> &#8290; </mo> <msup> <mi> &#8520; </mi> <mi> n </mi> </msup> <mo> &#8290; </mo> <msub> <semantics> <mrow> <mrow> <mi> &#952; </mi> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> P </mi> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[RowBox[List[InterpretationBox[&quot;\[Theta]&quot;, UnitStep, Rule[Editable, False], Rule[Selectable, False]], &quot;(&quot;, RowBox[List[&quot;1&quot;, &quot;-&quot;, SuperscriptBox[&quot;z&quot;, &quot;2&quot;]]], &quot;)&quot;]], &quot;P&quot;]], LegendreP] </annotation> </semantics> <mi> n </mi> </msub> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> , </mo> <mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <msub> <mi> j </mi> <mi> n </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <msqrt> <mfrac> <mi> &#960; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <mrow> <msub> <mi> J </mi> <mrow> <mi> n </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> z </mi> <mo> &#8712; </mo> <mi> &#8477; </mi> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> ErrorBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> &#8497; </ms> <ms> t </ms> </apply> <ms> [ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> n </ms> </apply> <ms> ( </ms> <ms> t </ms> <ms> ) </ms> </list> </apply> <ms> ] </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SqrtBox </ci> <apply> <ci> FractionBox </ci> <ms> &#960; </ms> <ms> 2 </ms> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <ms> &#8520; </ms> <ms> n </ms> </apply> <apply> <ci> SubscriptBox </ci> <apply> <ci> TagBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> InterpretationBox </ci> <ms> &#952; </ms> <ci> UnitStep </ci> <apply> <ci> Rule </ci> <ci> Editable </ci> <false /> </apply> <apply> <ci> Rule </ci> <ci> Selectable </ci> <false /> </apply> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> - </ms> <apply> <ci> SuperscriptBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <ms> P </ms> </list> </apply> <ci> LegendreP </ci> </apply> <ms> n </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> &#8712; </ms> <ms> &#8469; </ms> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> j </ms> <ms> n </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SqrtBox </ci> <apply> <ci> FractionBox </ci> <ms> &#960; </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> z </ms> </list> </apply> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> J </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> + </ms> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 2 </ms> </apply> </list> </apply> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> z </ms> <ms> &#8712; </ms> <ms> &#8477; </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["FourierTransform", "[", RowBox[List[RowBox[List[SubscriptBox["j", "n"], "[", "t_", "]"]], ",", "t_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SqrtBox[FractionBox["\[Pi]", "2"]], " ", SuperscriptBox["\[ImaginaryI]", "n"], " ", RowBox[List["UnitStep", "[", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], "]"]], " ", RowBox[List["LegendreP", "[", RowBox[List["n", ",", "z"]], "]"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List[RowBox[List[SubscriptBox["j", "n"], "[", "z", "]"]], "\[Equal]", RowBox[List[SqrtBox[FractionBox["\[Pi]", RowBox[List["2", " ", "z"]]]], " ", RowBox[List["BesselJ", "[", RowBox[List[RowBox[List["n", "+", FractionBox["1", "2"]]], ",", "z"]], "]"]]]]]], "&&", RowBox[List["z", "\[Element]", "Reals"]]]]]]]]]]










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





2002-12-18