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BesselK






Mathematica Notation

Traditional Notation









Bessel-Type Functions > BesselK[nu,z] > Specific values > Specialized values > For fixed z > Symbolic rational nu





http://functions.wolfram.com/03.04.03.0007.01









  


  










Input Form





BesselK[\[Nu], z] == (((-1)^(Abs[\[Nu]] - 1/3) 2^(Abs[\[Nu]] - 2/3) Pi Gamma[-(1/3)])/(z^Abs[\[Nu]] (3^(5/6) Gamma[1 - Abs[\[Nu]]]))) (3^(2/3) z^(2/3) AiryAiPrime[(3/2)^(2/3) z^(2/3)] Sum[((Abs[\[Nu]] - k - 4/3)!/(k! (Abs[\[Nu]] - 2 k - 4/3)! Pochhammer[4/3, k] Pochhammer[1 - Abs[\[Nu]], k])) (-(z^2/4))^k, {k, 0, Abs[\[Nu]] - 4/3}] - 2^(2/3) AiryAi[(3/2)^(2/3) z^(2/3)] Sum[((Abs[\[Nu]] - k - 1/3)!/(k! (Abs[\[Nu]] - 2 k - 1/3)! Pochhammer[1/3, k] Pochhammer[1 - Abs[\[Nu]], k])) (-(z^2/4))^k, {k, 0, Abs[\[Nu]] - 1/3}]) /; Element[Abs[\[Nu]] - 1/3, Integers]










Standard Form





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MathML Form







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</ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> k </ci> </apply> <apply> <factorial /> <apply> <plus /> <apply> <abs /> <ci> &#957; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> </apply> <apply> <ci> Pochhammer </ci> <cn type='rational'> 1 <sep /> 3 </cn> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <abs /> <ci> &#957; </ci> </apply> </apply> </apply> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <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> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <apply> <plus /> <apply> <abs /> <ci> &#957; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BesselK", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", FractionBox["1", "3"]]]], " ", SuperscriptBox["2", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", FractionBox["2", "3"]]]], " ", "\[Pi]", " ", SuperscriptBox["z", RowBox[List["-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]]], " ", RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["1", "3"]]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["z", RowBox[List["2", "/", "3"]]], " ", RowBox[List["AiryAiPrime", "[", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["z", RowBox[List["2", "/", "3"]]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", FractionBox["4", "3"]]]], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "k", "-", FractionBox["4", "3"]]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", RowBox[List["2", " ", "k"]], "-", FractionBox["4", "3"]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["4", "3"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ",", "k"]], "]"]]]]]]]]], "-", RowBox[List[SuperscriptBox["2", RowBox[List["2", "/", "3"]]], " ", RowBox[List["AiryAi", "[", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["z", RowBox[List["2", "/", "3"]]]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", FractionBox["1", "3"]]]], FractionBox[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", "k", "-", FractionBox["1", "3"]]], ")"]], "!"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]], ")"]], "k"]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", RowBox[List["2", " ", "k"]], "-", FractionBox["1", "3"]]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "3"], ",", "k"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], ",", "k"]], "]"]]]]]]]]]]], ")"]]]], RowBox[List[SuperscriptBox["3", RowBox[List["5", "/", "6"]]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", RowBox[List["Abs", "[", "\[Nu]", "]"]]]], "]"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "\[Nu]", "]"]], "-", FractionBox["1", "3"]]], "\[Element]", "Integers"]]]]]]]]










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





2001-10-29





© 1998- Wolfram Research, Inc.