Exponential Integral & Incomplete Gamma function

Pere Callahan

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/Incompl...aFunction.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

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

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