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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 0F~1





http://functions.wolfram.com/03.04.26.0034.01









  


  










Input Form





BesselK[\[Nu], z] Hypergeometric0F1Regularized[b, z^2/4] == (2^(-2 + b)/Sqrt[Pi]) MeijerG[{{(1 - b)/2, 1 - b/2}, {}}, {{-(\[Nu]/2), \[Nu]/2}, {1 - b - \[Nu]/2, 1 - b + \[Nu]/2}}, z^2] /; !(Element[-b - \[Nu], Integers] && -b - \[Nu] >= 0) && !(Element[-b + \[Nu], Integers] && -b + \[Nu] >= 0) && Inequality[-(Pi/2), Less, Arg[z], LessEqual, Pi/2]










Standard Form





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MathML Form







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</ci> </apply> <ci> z </ci> </apply> <apply> <ci> Hypergeometric0F1Regularized </ci> <ci> b </ci> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> b </ci> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </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> b </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> b </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </list> <list /> </list> <list> <list> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> &#957; 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</ci> </apply> </apply> <integers /> </apply> <apply> <notin /> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> </apply> <integers /> </apply> <apply> <ci> Inequality </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <lt /> <apply> <arg /> <ci> z </ci> </apply> <leq /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </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]_", ",", "z_"]], "]"]], " ", RowBox[List["Hypergeometric0F1Regularized", "[", RowBox[List["b_", ",", FractionBox[SuperscriptBox["z_", "2"], "4"]]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "+", "b"]]], " ", RowBox[List["MeijerG", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[FractionBox[RowBox[List["1", "-", "b"]], "2"], ",", RowBox[List["1", "-", FractionBox["b", "2"]]]]], "}"]], ",", RowBox[List["{", "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", FractionBox["\[Nu]", "2"]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["1", "-", "b", "-", FractionBox["\[Nu]", "2"]]], ",", RowBox[List["1", "-", "b", "+", FractionBox["\[Nu]", "2"]]]]], "}"]]]], "}"]], ",", SuperscriptBox["z", "2"]]], "]"]]]], SqrtBox["\[Pi]"]], "/;", RowBox[List[RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "b"]], "-", "\[Nu]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List["-", "b"]], "-", "\[Nu]"]], "\[GreaterEqual]", "0"]]]], ")"]]]], "&&", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "b"]], "+", "\[Nu]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List["-", "b"]], "+", "\[Nu]"]], "\[GreaterEqual]", "0"]]]], ")"]]]], "&&", RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "2"]]], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", FractionBox["\[Pi]", "2"]]]]]]]]]]]










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





2001-10-29