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BesselK






Mathematica Notation

Traditional Notation









Bessel-Type Functions > BesselK[nu,z] > Series representations > Generalized power series > Expansions on branch cuts > For the function itself





http://functions.wolfram.com/03.04.06.0025.01









  


  










Input Form





BesselK[\[Nu], z] \[Proportional] BesselK[\[Nu], x]/E^(2 I Pi \[Nu] Floor[Arg[-x + z]/(2 Pi)]) - 2 I Pi BesselI[\[Nu], x] Cos[Pi \[Nu]] Floor[Arg[-x + z]/(2 Pi)] - (1/2) ((BesselK[-1 + \[Nu], x] + BesselK[1 + \[Nu], x])/ E^(2 I Pi \[Nu] Floor[Arg[-x + z]/(2 Pi)]) + 2 I Pi (BesselI[-1 + \[Nu], x] + BesselI[1 + \[Nu], x]) Cos[Pi \[Nu]] Floor[Arg[-x + z]/(2 Pi)]) (z - x) + (1/8) ((BesselK[-2 + \[Nu], x] + 2 BesselK[\[Nu], x] + BesselK[2 + \[Nu], x])/E^(2 I Pi \[Nu] Floor[Arg[-x + z]/(2 Pi)]) - 2 I Pi (BesselI[-2 + \[Nu], x] + 2 BesselI[\[Nu], x] + BesselI[2 + \[Nu], x]) Cos[Pi \[Nu]] Floor[Arg[-x + z]/(2 Pi)]) (z - x)^2 + O[(z - x)^3] /; Element[x, Reals] && x < 0










Standard Form





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MathML Form







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</mo> <mrow> <mrow> <mi> x </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[List[], Reals]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> x </mi> <mo> &lt; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> BesselK </ci> <ci> &#957; </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -2 </cn> <imaginaryi /> <pi /> <ci> &#957; </ci> <apply> <floor /> <apply> <times /> <apply> <arg /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> BesselK </ci> <ci> &#957; 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</ci> <ci> x </ci> </apply> </apply> <apply> <ci> BesselI </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> <ci> x </ci> </apply> </apply> <apply> <cos /> <apply> <times /> <pi /> <ci> &#957; </ci> </apply> </apply> <apply> <floor /> <apply> <times /> <apply> <arg /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> x </ci> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> x </ci> <reals /> </apply> <apply> <lt /> <ci> x </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2007-05-02





© 1998- Wolfram Research, Inc.