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BesselK






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/03.04.03.0020.01









  


  










Input Form





BesselK[13/3, z] == (1/(27 2^(1/3) 3^(5/6) z^(13/3))) (Pi (2^(2/3) (4480 + 3024 z^2 + 81 z^4) AiryAi[(3/2)^(2/3) z^(2/3)] - 56 3^(2/3) z^(2/3) (80 + 9 z^2) AiryAiPrime[(3/2)^(2/3) z^(2/3)]))










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["BesselK", "[", RowBox[List[FractionBox["13", "3"], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["27", " ", SuperscriptBox["2", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox["3", RowBox[List["5", "/", "6"]]], " ", SuperscriptBox["z", RowBox[List["13", "/", "3"]]]]]], RowBox[List["(", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["2", "/", "3"]]], " ", RowBox[List["(", RowBox[List["4480", "+", RowBox[List["3024", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["81", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["AiryAi", "[", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["z", RowBox[List["2", "/", "3"]]]]], "]"]]]], "-", RowBox[List["56", " ", SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["z", RowBox[List["2", "/", "3"]]], " ", RowBox[List["(", RowBox[List["80", "+", RowBox[List["9", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["AiryAiPrime", "[", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["z", RowBox[List["2", "/", "3"]]]]], "]"]]]]]], ")"]]]], ")"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msub> <mi> K </mi> <mfrac> <mn> 13 </mn> <mn> 3 </mn> </mfrac> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mi> &#960; </mi> <mtext> </mtext> </mrow> <mrow> <mn> 27 </mn> <mo> &#8290; </mo> <mroot> <mn> 2 </mn> <mn> 3 </mn> </mroot> <mo> &#8290; </mo> <msup> <mn> 3 </mn> <mrow> <mn> 5 </mn> <mo> / </mo> <mn> 6 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 13 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mn> 2 </mn> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 81 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3024 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 4480 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> Ai </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 56 </mn> <mo> &#8290; </mo> <msup> <mn> 3 </mn> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 9 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mn> 80 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> Ai </mi> <mo> &#8242; </mo> </msup> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> BesselK </ci> <cn type='rational'> 13 <sep /> 3 </cn> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <pi /> <apply> <power /> <apply> <times /> <cn type='integer'> 27 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 5 <sep /> 6 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 13 <sep /> 3 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 81 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3024 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 4480 </cn> </apply> <apply> <ci> AiryAi </ci> <apply> <times /> <apply> <power /> <cn type='rational'> 3 <sep /> 2 </cn> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 2 <sep /> 3 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 56 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 9 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> 80 </cn> </apply> <apply> <ci> AiryAiPrime </ci> <apply> <times /> <apply> <power /> <cn type='rational'> 3 <sep /> 2 </cn> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 2 <sep /> 3 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["BesselK", "[", RowBox[List[FractionBox["13", "3"], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["2", "/", "3"]]], " ", RowBox[List["(", RowBox[List["4480", "+", RowBox[List["3024", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["81", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["AiryAi", "[", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["z", RowBox[List["2", "/", "3"]]]]], "]"]]]], "-", RowBox[List["56", " ", SuperscriptBox["3", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["z", RowBox[List["2", "/", "3"]]], " ", RowBox[List["(", RowBox[List["80", "+", RowBox[List["9", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["AiryAiPrime", "[", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], RowBox[List["2", "/", "3"]]], " ", SuperscriptBox["z", RowBox[List["2", "/", "3"]]]]], "]"]]]]]], ")"]]]], RowBox[List["27", " ", SuperscriptBox["2", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox["3", RowBox[List["5", "/", "6"]]], " ", SuperscriptBox["z", RowBox[List["13", "/", "3"]]]]]]]]]]










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





2002-12-18





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