Bessel function of the first kind: Transformations (formula 03.01.16.0003)

BesselJ

$Mathematica Notation$

Bessel-Type Functions

BesselJ[nu,z]

Transformations

Transformations and argument simplifications

Argument involving basic arithmetic operations

http://functions.wolfram.com/03.01.16.0003.01

Input Form

BesselJ[\[Nu], (-I) z] == (((-I) z)^\[Nu] BesselI[\[Nu], z])/z^\[Nu]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]]], "]"]], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]], ")"]], "\[Nu]"], " ", RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], SuperscriptBox["z", "\[Nu]"]]]]]]

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> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mi> ν </mi> </msup> <msup> <mi> z </mi> <mi> ν </mi> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <msub> <mi> I </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> BesselJ </ci> <ci> ν </ci> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> z </ci> </apply> <ci> ν </ci> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <ci> ν </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> BesselI </ci> <ci> ν </ci> <ci> z </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]_", ",", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z_"]]]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]], ")"]], "\[Nu]"], " ", RowBox[List["BesselI", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], SuperscriptBox["z", "\[Nu]"]]]]]]

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

2001-10-29