Kelvin function of the first kind: Representations through equivalent functions (formula 03.17.27.0007)

KelvinBei

$Mathematica Notation$

Bessel-Type Functions

KelvinBei[nu,z]

Representations through equivalent functions

With related functions

http://functions.wolfram.com/03.17.27.0007.01

Input Form

KelvinBei[\[Nu], z] + I KelvinBer[\[Nu], z] == Piecewise[{{I E^(I Pi \[Nu]) BesselJ[\[Nu], (-1)^(1/4) z], Inequality[(3 Pi)/4, Less, Arg[z], LessEqual, Pi]}}, (I BesselJ[\[Nu], (-1)^(1/4) z])/E^(I Pi \[Nu])]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["KelvinBei", "[", RowBox[List["\[Nu]", ",", " ", "z"]], "]"]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["KelvinBer", "[", RowBox[List["\[Nu]", ",", " ", "z"]], "]"]]]]]], "\[Equal]", RowBox[List["Piecewise", "[", RowBox[List[RowBox[List["{", RowBox[List["{", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]], " ", RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "z"]]]], "]"]]]], ",", RowBox[List[FractionBox[RowBox[List["3", "\[Pi]"]], "4"], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", "\[Pi]"]]]], "}"]], "}"]], ",", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "\[Pi]", " ", "\[Nu]"]]], RowBox[List["BesselJ", "[", RowBox[List["\[Nu]", ",", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["1", "/", "4"]]], " ", "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> <mrow> <msub> <mi> bei </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <msub> <mi> ber </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo>  </mo> <mrow> <mo>  </mo> <mtable> <mtr> <mtd> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msub> <mi> J </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mrow> <mroot> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </mroot> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mfrac> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mn> 4 </mn> </mfrac> <mo> < </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ≤ </mo> <mi> π </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <msub> <mi> J </mi> <mi> ν </mi> </msub> <mo> ( </mo> <mrow> <mroot> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </mroot> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mtd> <mtd> <semantics> <mi> True </mi> <annotation encoding='Mathematica'> TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]] </annotation> </semantics> </mtd> </mtr> </mtable> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <plus /> <apply> <ci> KelvinBei </ci> <ci> ν </ci> <ci> z </ci> </apply> <apply> <times /> <imaginaryi /> <apply> <ci> KelvinBer </ci> <ci> ν </ci> <ci> z </ci> </apply> </apply> </apply> <piecewise> <piece> <apply> <times /> <imaginaryi /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> ν </ci> </apply> </apply> <apply> <ci> BesselJ </ci> <ci> ν </ci> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <ci> Inequality </ci> <apply> <times /> <cn type='integer'> 3 </cn> <pi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <lt /> <apply> <arg /> <ci> z </ci> </apply> <leq /> <pi /> </apply> </piece> <otherwise> <apply> <times /> <imaginaryi /> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <pi /> <ci> ν </ci> </apply> </apply> <apply> <ci> BesselJ </ci> <ci> ν </ci> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> </otherwise> </piecewise> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2007-05-02

KelvinBei[z]