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BesselI






Mathematica Notation

Traditional Notation









Bessel-Type Functions > BesselI[nu,z] > Series representations > Asymptotic series expansions > Expansions for any z in trigonometric form > Using trigonometric functions with branch cut-free arguments





http://functions.wolfram.com/03.02.06.0082.01









  


  










Input Form





BesselI[\[Nu], z] \[Proportional] (z^\[Nu]/((I z)^\[Nu] Sqrt[2 Pi])) (((E^(I Pi \[Nu])/Sqrt[(-I) z]) (1 + Sqrt[z^2]/z) Cosh[z - (I Pi (1 + 2 \[Nu]))/4] + (1/Sqrt[I z]) (1 - Sqrt[z^2]/z) Cosh[z + (I Pi (1 + 2 \[Nu]))/4]) (1 + (9 - 40 \[Nu]^2 + 16 \[Nu]^4)/ (128 z^2) + (11025 - 51664 \[Nu]^2 + 31584 \[Nu]^4 - 5376 \[Nu]^6 + 256 \[Nu]^8)/(98304 z^4) + \[Ellipsis]) + ((1 - 4 \[Nu]^2)/(8 z)) ((E^(I Pi \[Nu])/Sqrt[(-I) z]) (1 + Sqrt[z^2]/z) Sinh[z - (I Pi (1 + 2 \[Nu]))/4] + (1/Sqrt[I z]) (1 - Sqrt[z^2]/z) Sinh[z + (I Pi (1 + 2 \[Nu]))/4]) (1 + (225 - 136 \[Nu]^2 + 16 \[Nu]^4)/(384 z^2) + (893025 - 656784 \[Nu]^2 + 137824 \[Nu]^4 - 10496 \[Nu]^6 + 256 \[Nu]^8)/(491520 z^4) + \[Ellipsis])) /; (Abs[z] -> Infinity)










Standard Form





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MathML Form







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</ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 31584 </cn> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 5376 </cn> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 256 </cn> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 8 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 98304 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 8 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <sinh /> <apply> <plus /> <ci> z </ci> <apply> <times /> <imaginaryi /> <pi /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <ci> &#957; </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <sinh /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <pi /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> &#957; </ci> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <cn type='integer'> 225 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 136 </cn> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 16 </cn> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 384 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <cn type='integer'> 893025 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 656784 </cn> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 137824 </cn> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 10496 </cn> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 256 </cn> <apply> <power /> <ci> &#957; </ci> <cn type='integer'> 8 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 491520 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <apply> <abs /> <ci> z </ci> </apply> <infinity /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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





2007-05-02