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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/03.04.26.0085.01









  


  










Input Form





BesselK[\[Nu], 2 Sqrt[z]] Hypergeometric0F1[-n - \[Nu], z] == 2^(-(3/2) - n) Sqrt[Pi] z^(\[Nu]/2) Gamma[-n - \[Nu]] ((Pi Csc[(1/4) Pi (-(-1)^n + 4 \[Nu])] MeijerG[{{(1/2) (1 + n + \[Nu]), (1/2) (2 + n + \[Nu])}, {(1/4) (1 - 2 n - 2 \[Nu]), (1/4) (3 + 2 n + 2 \[Nu])}}, {{1 + n + \[Nu]/2, -(\[Nu]/2)}, {(1/4) (1 - 2 n - 2 \[Nu]), (1/4) (3 + 2 n + 2 \[Nu]), \[Nu]/2, 1 + n + (3 \[Nu])/2}}, 4 z])/ (2^\[Nu] z^(\[Nu]/2)) - Sqrt[2] Csc[\[Nu] Pi] Sum[(((-1)^Floor[(1 + n)/2] 4^k z^k Gamma[1/2 + k - n + Floor[n/2]])/ (k! Gamma[1 + k + \[Nu]] Gamma[k - n - \[Nu]])) Pochhammer[1 - k + Floor[n/2], n - Floor[n/2]], {k, 0, Floor[n/2]}]) /; Element[n, Integers] && n >= 0










Standard Form





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MathML Form







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</mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;1&quot;, &quot;-&quot;, &quot;k&quot;, &quot;+&quot;, RowBox[List[&quot;\[LeftFloor]&quot;, FractionBox[&quot;n&quot;, &quot;2&quot;], &quot;\[RightFloor]&quot;]]]], &quot;)&quot;]], RowBox[List[&quot;n&quot;, &quot;-&quot;, RowBox[List[&quot;\[LeftFloor]&quot;, FractionBox[&quot;n&quot;, &quot;2&quot;], &quot;\[RightFloor]&quot;]]]]], Pochhammer] </annotation> </semantics> </mrow> <mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> &#957; </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mi> n </mi> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <ci> BesselK </ci> <ci> &#957; </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> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <ci> &#957; 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Rule Form





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





2007-05-02