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variants of this functions
KelvinBer




Mathematica Notation

Traditional Notation









Bessel-Type Functions > KelvinBer[nu,z] > Integral transforms > Laplace transforms





http://functions.wolfram.com/03.18.22.0001.01









  


  








Input Form





LaplaceTransform[KelvinBer[\[Nu], t], t, z] == 2^(-2 - \[Nu]) z^(-3 - \[Nu]) (4 z^2 Cos[(3 Pi \[Nu])/4] HypergeometricPFQ[{1/4 + \[Nu]/4, 1/2 + \[Nu]/4, 3/4 + \[Nu]/4, 1 + \[Nu]/4}, {1/2, 1/2 + \[Nu]/2, 1 + \[Nu]/2}, -(1/z^4)] - (2 + \[Nu]) Sin[(3 Pi \[Nu])/4] HypergeometricPFQ[{3/4 + \[Nu]/4, 1 + \[Nu]/4, 5/4 + \[Nu]/4, 3/2 + \[Nu]/4}, {3/2, 1 + \[Nu]/2, 3/2 + \[Nu]/2}, -(1/z^4)]) /; Re[\[Nu]] > -1 && Re[z] > 1/Sqrt[2]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LaplaceTransform", "[", RowBox[List[RowBox[List["KelvinBer", "[", RowBox[List["\[Nu]", ",", "t"]], "]"]], ",", "t", ",", "z"]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "\[Nu]"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "3"]], "-", "\[Nu]"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["z", "2"], " ", RowBox[List["Cos", "[", FractionBox[RowBox[List["3", " ", "\[Pi]", " ", "\[Nu]"]], "4"], "]"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], "+", FractionBox["\[Nu]", "4"]]], ",", RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Nu]", "4"]]], ",", RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Nu]", "4"]]], ",", RowBox[List["1", "+", FractionBox["\[Nu]", "4"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List[FractionBox["1", "2"], "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "4"]]]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]], " ", RowBox[List["Sin", "[", FractionBox[RowBox[List["3", " ", "\[Pi]", " ", "\[Nu]"]], "4"], "]"]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[FractionBox["3", "4"], "+", FractionBox["\[Nu]", "4"]]], ",", RowBox[List["1", "+", FractionBox["\[Nu]", "4"]]], ",", RowBox[List[FractionBox["5", "4"], "+", FractionBox["\[Nu]", "4"]]], ",", RowBox[List[FractionBox["3", "2"], "+", FractionBox["\[Nu]", "4"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["3", "2"], ",", RowBox[List["1", "+", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["3", "2"], "+", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["-", FractionBox["1", SuperscriptBox["z", "4"]]]]]], "]"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Re", "[", "\[Nu]", "]"]], ">", RowBox[List["-", "1"]]]], "\[And]", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", FractionBox["1", SqrtBox["2"]]]]]]]]]]








MathML Form







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</mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 4 </mn> </msub> <msub> <mi> F </mi> <mn> 3 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mfrac> <mi> &#957; </mi> <mn> 4 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mi> &#957; </mi> <mn> 4 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mfrac> <mi> &#957; </mi> <mn> 4 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 5 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mi> &#957; </mi> <mn> 4 </mn> </mfrac> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mfrac> <mi> &#957; 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</mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#957; </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mfrac> <mn> 1 </mn> <msqrt> <mn> 2 </mn> </msqrt> </mfrac> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> LaplaceTransform </ci> <apply> <ci> KelvinBer </ci> <ci> &#957; </ci> <ci> t </ci> </apply> <ci> t </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> -3 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <cos /> <apply> <times /> <cn type='integer'> 3 </cn> <pi /> <ci> &#957; 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Rule Form





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





2007-05-02