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BesselK






Mathematica Notation

Traditional Notation









Bessel-Type Functions > BesselK[nu,z] > Representations through more general functions > Through Meijer G > Classical cases involving 0F1





http://functions.wolfram.com/03.04.26.0086.01









  


  










Input Form





BesselK[\[Nu], 2 Sqrt[z]] Hypergeometric0F1[\[Nu], z] == (Gamma[\[Nu]]/2) (z^(-\[Nu]/2) - 2^(-(1/2) + \[Nu]) Pi^(3/2) Csc[(1/4) (1 + 4 \[Nu]) Pi] MeijerG[{{(1 - \[Nu])/2, 1 - \[Nu]/2}, {(3 - 2 \[Nu])/4, (1 + 2 \[Nu])/4}}, {{1 - \[Nu]/2, \[Nu]/2}, {(3 - 2 \[Nu])/4, 1 - (3 \[Nu])/2, -(\[Nu]/2), (1 + 2 \[Nu])/4}}, 4 z])










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["BesselK", "[", RowBox[List["\[Nu]", ",", RowBox[List["2", SqrtBox["z"]]]]], "]"]], " ", RowBox[List["Hypergeometric0F1", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["Gamma", "[", "\[Nu]", "]"]], "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List[RowBox[List["-", "\[Nu]"]], "/", "2"]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "\[Nu]"]]], " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", RowBox[List["Csc", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]], " ", "\[Pi]"]], "]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["3", "-", RowBox[List["2", "\[Nu]"]]]], "4"], " ", ",", FractionBox[RowBox[List["1", "+", RowBox[List["2", "\[Nu]"]]]], "4"]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]], ",", FractionBox["\[Nu]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List["3", "-", RowBox[List["2", "\[Nu]"]]]], "4"], ",", RowBox[List["1", "-", FractionBox[RowBox[List["3", " ", "\[Nu]"]], "2"]]], ",", RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", FractionBox[RowBox[List["1", "+", RowBox[List["2", "\[Nu]"]]]], "4"]]], " ", "}"]]]], "}"]], ",", RowBox[List["4", "z"]]]], "]"]]]]]], ")"]]]]]]]]










MathML Form







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</ci> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <ci> Hypergeometric0F1 </ci> <ci> &#957; </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <ci> Gamma </ci> <ci> &#957; </ci> </apply> <apply> <plus /> <apply> <power /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <csc /> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 1 </cn> </apply> <pi /> </apply> </apply> <apply> <ci> MeijerG </ci> <list> <list> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </list> <list> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <plus /> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </list> </list> <list> <list> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </list> <list> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <plus /> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </list> </list> <apply> <times /> <cn type='integer'> 4 </cn> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["BesselK", "[", RowBox[List["\[Nu]_", ",", RowBox[List["2", " ", SqrtBox["z_"]]]]], "]"]], " ", RowBox[List["Hypergeometric0F1", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Gamma", "[", "\[Nu]", "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["z", RowBox[List["-", FractionBox["\[Nu]", "2"]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "+", "\[Nu]"]]], " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]], " ", RowBox[List["Csc", "[", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", "\[Nu]"]]]], ")"]], " ", "\[Pi]"]], "]"]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["1", "-", FractionBox["\[Nu]", "2"]]], ",", FractionBox["\[Nu]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]], ",", RowBox[List["1", "-", FractionBox[RowBox[List["3", " ", "\[Nu]"]], "2"]]], ",", RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", "\[Nu]"]]]], ")"]]]]]], "}"]]]], "}"]], ",", RowBox[List["4", " ", "z"]]]], "]"]]]]]], ")"]]]]]]]]










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





2007-05-02





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