Kelvin function of the first kind: Representations through more general functions (formula 03.18.26.0068)

KelvinBer

$Mathematica Notation$

Bessel-Type Functions

KelvinBer[nu,z]

Representations through more general functions

Through Meijer G

Generalized cases involving bei

http://functions.wolfram.com/03.18.26.0068.01

Input Form

KelvinBer[\[Mu], z] KelvinBer[\[Nu], z] - KelvinBei[\[Nu], z] KelvinBei[\[Mu], z] == 2^(3/2) Pi^(5/2) MeijerG[{{0, 1/4, 3/4}, {1/6, 1/2, 5/6}}, {{(\[Mu] + \[Nu])/4, (\[Mu] + \[Nu] + 2)/4}, {1/6, 5/6, -((\[Mu] + \[Nu])/4), (\[Mu] - \[Nu])/4, (\[Nu] - \[Mu])/4, (2 - \[Mu] - \[Nu])/4, (\[Mu] - \[Nu] + 2)/4, (\[Nu] - \[Mu] + 2)/4}}, z/2, 1/4]

Standard Form

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MathML Form

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⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mfrac> <mi> z </mi> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ❘ </mo> <mtable> <mtr> <mtd> <mrow> <mn> 0 </mn> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 5 </mn> <mn> 6 </mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mfrac> <mrow> <mi> μ </mi> <mo> + </mo> <mi> ν </mi> </mrow> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mrow> <mi> μ </mi> <mo> + </mo> <mi> ν </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mn> 4 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 1 </mn> <mn> 6 </mn> </mfrac> <mo> , </mo> <mfrac> <mn> 5 </mn> <mn> 6 </mn> </mfrac> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> μ </mi> <mo> + </mo> <mi> ν </mi> </mrow> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> <mfrac> <mrow> <mi> 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1 <sep /> 4 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List[RowBox[List["KelvinBer", "[", RowBox[List["\[Mu]_", ",", "z_"]], "]"]], " ", RowBox[List["KelvinBer", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "-", RowBox[List[RowBox[List["KelvinBei", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], " ", RowBox[List["KelvinBei", "[", RowBox[List["\[Mu]_", ",", "z_"]], "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox["2", RowBox[List["3", "/", "2"]]], " ", SuperscriptBox["\[Pi]", RowBox[List["5", "/", "2"]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", FractionBox["1", "4"], ",", FractionBox["3", "4"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "6"], ",", FractionBox["1", "2"], ",", FractionBox["5", "6"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["\[Mu]", "+", "\[Nu]"]], "4"], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]", "+", "2"]], ")"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "6"], ",", FractionBox["5", "6"], ",", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["\[Mu]", "+", "\[Nu]"]], ")"]]]], ",", FractionBox[RowBox[List["\[Mu]", "-", "\[Nu]"]], "4"], ",", FractionBox[RowBox[List["\[Nu]", "-", "\[Mu]"]], "4"], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["2", "-", "\[Mu]", "-", "\[Nu]"]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["\[Mu]", "-", "\[Nu]", "+", "2"]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["\[Nu]", "-", "\[Mu]", "+", "2"]], ")"]]]]]], "}"]]]], "}"]], ",", FractionBox["z", "2"], ",", FractionBox["1", "4"]]], "]"]]]]]]]]

Contributed by

Brychkov Yu.A. (2006)

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

2007-05-02

KelvinBer[z]