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variants of this functions
KelvinBei




Mathematica Notation

Traditional Notation









Bessel-Type Functions > KelvinBei[nu,z] > Series representations > Asymptotic series expansions > Expansions inside Stokes sectors > Expansions containing z->infinity > In exponential form ||| In exponential form





http://functions.wolfram.com/03.17.06.0034.01









  


  








Input Form





KelvinBei[\[Nu], z] \[Proportional] (-(1/(2 Sqrt[2 Pi] Sqrt[z]))) (Sum[((Pochhammer[1/2 - \[Nu], 2 k] Pochhammer[1/2 + \[Nu], 2 k])/(2 k)!) (I/(4 z^2))^k (E^(z/Sqrt[2]) (E^(-((I z)/Sqrt[2]) - (I Pi \[Nu])/2 - (3 Pi I)/8) + (-1)^k E^((I z)/Sqrt[2] + (I Pi \[Nu])/2 + (3 Pi I)/8)) + ((-(-1)^k) E^(-((I z)/Sqrt[2]) + (3 I Pi \[Nu])/2 - (Pi I)/8) + E^((I z)/Sqrt[2] + (I Pi \[Nu])/2 + (Pi I)/8))/E^(z/Sqrt[2])), {k, 0, Floor[n/2]}] + (1/(2 z)) Sum[((Pochhammer[1/2 - \[Nu], 1 + 2 k] Pochhammer[1/2 + \[Nu], 1 + 2 k])/ (1 + 2 k)!) (I/(4 z^2))^k (E^(z/Sqrt[2]) (E^(-((I z)/Sqrt[2]) - (I Pi \[Nu])/2 - (Pi I)/8) + (-1)^k E^((I z)/Sqrt[2] + (I Pi \[Nu])/2 + (Pi I)/8)) + ((-1)^k E^(-((I z)/Sqrt[2]) + (3 I Pi \[Nu])/2 - (3 Pi I)/8) - E^((I z)/Sqrt[2] + (I Pi \[Nu])/2 + (3 Pi I)/8))/E^(z/Sqrt[2])), {k, 0, Floor[(n - 1)/2]}] + \[Ellipsis]) /; Inequality[-(Pi/2), Less, Arg[z], LessEqual, Pi] && (Abs[z] -> Infinity) && Element[n, Integers] && n >= 0








Standard Form





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MathML Form







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</ci> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <apply> <ci> Pochhammer </ci> <apply> <plus /> <ci> &#957; </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <factorial /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> k </ci> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <imaginaryi /> <pi /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 8 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <imaginaryi /> <pi /> <ci> &#957; </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 8 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <pi /> <ci> &#957; 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Rule Form





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





2007-05-02