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

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