Exponential Integral & Incomplete Gamma function

[Pere Callahan]

Hello,

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

I am using the notation from Mathworld here

http://mathworld.wolfram.com/ExponentialIntegral.html
http://mathworld.wolfram.com/IncompleteGammaFunction.htmlI am interested in the behaviour of their difference as k\to\infty. 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

 

[Pere Callahan]

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

 

[Pere Callahan]

Ok. Solved. I bound the difference by

\frac{e^m}{2k+1}