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Exponential Integral & Incomplete Gamma function

  1. Jan 16, 2008 #1
    Hello,

    I need to compare an exponential integral [tex]-E_{-2k}(-m)[/tex] -where k is a positive integer and m just a real number- to a Gamma function [tex]\frac{1}{m^{2k+1}}\Gamma(2k+1)[/tex].

    I am using the notation from Mathworld here

    http://mathworld.wolfram.com/ExponentialIntegral.html
    http://mathworld.wolfram.com/IncompleteGammaFunction.html


    I am interested in the behaviour of their difference as [tex]k\to\infty[/tex]. It seems to tend to zero, but are there any estimates as to how fast the difference goes to zero?

    Thansk for any comments.


    -Pere
     
    Last edited: Jan 17, 2008
  2. jcsd
  3. Jan 17, 2008 #2
    Maybe I ask should another question first.

    How is the exponential integral function defined for real z less than zero....the integral representation clearly does not converge in that case... is it just analytic continuation or is there an explicit formula...?

    Thanks

    -Pere
     
  4. Jan 18, 2008 #3
    Ok. Solved. I bound the difference by

    [tex]\frac{e^m}{2k+1}[/tex]
     
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