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BesselK




Mathematica Notation

Traditional Notation









Bessel-Type Functions > BesselK[nu,z] > Series representations > Generalized power series > Expansions at z==0 > For the function itself > Logarithmic cases





http://functions.wolfram.com/03.04.06.0006.01









  


  








Input Form





BesselK[n, z] == ((-1)^n/8) ((-((2^(4 - n) z^(-2 + n))/(n - 1)!)) HypergeometricPFQ[{1, 1, 1 - n}, {}, 4/z^2] + (z^(2 + n)/(2^n ((1 + n) (1 + n)!))) ((1 + n) HypergeometricPFQ[{{}, {1}, {1, 1}}, {{2, 2 + n}, {}, {2}}, z^2/4, z^2/4] + HypergeometricPFQ[{{}, {1}, {1, 1 + n}}, {{2, 2 + n}, {}, {2 + n}}, z^2/4, z^2/4]) - 4 BesselI[n, z] (EulerGamma + 2 Log[z/2] - PolyGamma[1 + n])) /; Element[n, Integers] && n >= 0








Standard Form





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MathML Form







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</ms> </list> </apply> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <ms> F </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> 0 </ms> <ms> 1 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 0 </ms> <ms> 1 </ms> <ms> 2 </ms> </list> </apply> </apply> <ms> [ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> GridBox </ci> <list> <list> <apply> <ci> ErrorBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ; </ms> <ms> 1 </ms> <ms> ; </ms> <ms> 1 </ms> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <ms> 1 </ms> <ms> ; </ms> </list> </apply> </list> </apply> </apply> </list> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> + </ms> <ms> 2 </ms> </list> </apply> <ms> ; </ms> </list> </apply> <ms> ; </ms> <ms> 2 </ms> <ms> ; </ms> </list> </apply> </list> </apply> </list> </list> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> SuperscriptBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> <ms> 4 </ms> </apply> </list> </apply> <ms> , </ms> <apply> <ci> FractionBox </ci> <apply> <ci> SuperscriptBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> <ms> 4 </ms> </apply> <ms> ] </ms> </list> </apply> </list> </apply> <ms> + </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubsuperscriptBox </ci> <ms> F </ms> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> 0 </ms> <ms> 1 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> 0 </ms> <ms> 1 </ms> <ms> 2 </ms> </list> </apply> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> GridBox </ci> <list> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ; </ms> <ms> 1 </ms> <ms> ; </ms> <ms> 1 </ms> </list> </apply> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ; </ms> </list> </apply> </list> </apply> </list> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <ms> , </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> + </ms> <ms> 2 </ms> </list> </apply> <ms> ; </ms> </list> </apply> <ms> ; </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> + </ms> <ms> 2 </ms> </list> </apply> <ms> ; </ms> </list> </apply> </list> </apply> </list> </list> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> SuperscriptBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> <ms> 4 </ms> </apply> </list> </apply> <ms> , </ms> <apply> <ci> FractionBox </ci> <apply> <ci> SuperscriptBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> <ms> 4 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <ms> 4 </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> I </ms> <ms> n </ms> </apply> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> RowBox </ci> <list> <ms> log </ms> <ms> ( </ms> <apply> <ci> FractionBox </ci> <ms> z </ms> <ms> 2 </ms> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> - </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> TagBox </ci> <ms> &#968; </ms> <ci> PolyGamma </ci> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <ms> + </ms> <apply> <ci> TagBox </ci> <ms> &#8509; </ms> <apply> <ci> Function </ci> <eulergamma /> </apply> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> &#8712; </ms> <apply> <ci> TagBox </ci> <ms> &#8469; </ms> <apply> <ci> Function </ci> <integers /> </apply> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BesselK", "[", RowBox[List["n_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "8"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["4", "-", "n"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "2"]], "+", "n"]]]]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", "1", ",", RowBox[List["1", "-", "n"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", FractionBox["4", SuperscriptBox["z", "2"]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "1"]], ")"]], "!"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "n"]]], " ", SuperscriptBox["z", RowBox[List["2", "+", "n"]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", "1"]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List["2", "+", "n"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", "2", "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "4"], ",", FractionBox[SuperscriptBox["z", "2"], "4"]]], "]"]]]], "+", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", "}"]], ",", RowBox[List["{", "1", "}"]], ",", RowBox[List["{", RowBox[List["1", ",", RowBox[List["1", "+", "n"]]]], "}"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["2", ",", RowBox[List["2", "+", "n"]]]], "}"]], ",", RowBox[List["{", "}"]], ",", RowBox[List["{", RowBox[List["2", "+", "n"]], "}"]]]], "}"]], ",", FractionBox[SuperscriptBox["z", "2"], "4"], ",", FractionBox[SuperscriptBox["z", "2"], "4"]]], "]"]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]], " ", RowBox[List[RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]], "!"]]]]], "-", RowBox[List["4", " ", RowBox[List["BesselI", "[", RowBox[List["n", ",", "z"]], "]"]], " ", RowBox[List["(", RowBox[List["EulerGamma", "+", RowBox[List["2", " ", RowBox[List["Log", "[", FractionBox["z", "2"], "]"]]]], "-", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "n"]], "]"]]]], ")"]]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]


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





2001-10-29