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variants of this functions
HermiteH






Mathematica Notation

Traditional Notation









Hypergeometric Functions > HermiteH[nu,z] > Series representations > Asymptotic series expansions





http://functions.wolfram.com/07.01.06.0025.01









  


  










Input Form





HermiteH[\[Nu], z] \[Proportional] Piecewise[{{2^\[Nu] z^\[Nu] (1 - ((-1 + \[Nu]) \[Nu])/(4 z^2) + ((-3 + \[Nu]) (-2 + \[Nu]) (-1 + \[Nu]) \[Nu])/(32 z^4) + O[z^(-6)]), Inequality[-(Pi/2), Less, Arg[z], LessEqual, Pi/2]}, {2^\[Nu] z^\[Nu] (1 - ((-1 + \[Nu]) \[Nu])/(4 z^2) + ((-3 + \[Nu]) (-2 + \[Nu]) (-1 + \[Nu]) \[Nu])/(32 z^4) + O[z^(-6)]) - ((E^(z^2 + I Pi \[Nu]) Sqrt[Pi] z^(-1 - \[Nu]))/ Gamma[-\[Nu]]) (1 + ((1 + \[Nu]) (2 + \[Nu]))/(4 z^2) + ((1 + \[Nu]) (2 + \[Nu]) (3 + \[Nu]) (4 + \[Nu]))/(32 z^4) + O[z^(-6)]), Arg[z] > Pi/2}}, 2^\[Nu] z^\[Nu] (1 - ((-1 + \[Nu]) \[Nu])/(4 z^2) + ((-3 + \[Nu]) (-2 + \[Nu]) (-1 + \[Nu]) \[Nu])/(32 z^4) + O[z^(-6)]) - ((E^(z^2 - I Pi \[Nu]) Sqrt[Pi] z^(-1 - \[Nu]))/Gamma[-\[Nu]]) (1 + ((1 + \[Nu]) (2 + \[Nu]))/(4 z^2) + ((1 + \[Nu]) (2 + \[Nu]) (3 + \[Nu]) (4 + \[Nu]))/(32 z^4) + O[z^(-6)])] /; (Abs[z] -> Infinity)










Standard Form





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MathML Form







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</ci> </apply> <apply> <power /> <ci> z </ci> <ci> &#957; </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> -1 </cn> </apply> <ci> &#957; </ci> <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> </apply> <apply> <times /> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> -3 </cn> </apply> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> -2 </cn> </apply> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> -1 </cn> </apply> <ci> &#957; </ci> <apply> <power /> <apply> <times /> <cn type='integer'> 32 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <imaginaryi /> <pi /> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> <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> <apply> <times /> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 3 </cn> </apply> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 4 </cn> </apply> <apply> <plus /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 32 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <gt /> <apply> <arg /> <ci> z </ci> </apply> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </piece> <otherwise> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> <apply> <power /> <ci> z </ci> <ci> &#957; </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> &#957; 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Rule Form





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





2007-05-02