From: Paul Abbott <paul@physics.uwa.edu.au>
Subject: Re: Like incomplete gamma computation
Date: Wed, 03 Feb 1999 14:40:15 +0800
Newsgroups: sci.math.num-analysis
To: Hideo HIROSE <hirose@ces.kyutech.ac.jp>
Keywords: Meijer functions

Hideo HIROSE wrote:

> I need some efficient and accurate methods to compute the following
> function:
> (1)  Integral_0 ^x [ log(u) u^(t-1) exp(-u) du
> (2)  Integral_0 ^x [ (log(u))^2 u^(t-1) exp(-u) du
>
>It looks like incomplete gamma function.
>Does anybody know the algorithm or related references?

Mathematica can compute these integrals in closed form -- in terms of
Meijer G functions.  However, observe that, using parametric
differentiation with respect to t, (1) and (2) are the first and second
derivatives of the base formula

	In[1]:= Integrate[u^(t - 1) Exp[-u], {u, 0, x}]
	Out[1]= Gamma[t] - Gamma[t, x]

where x > 0 and Re[t] > 0, and Gamma[t, x] is the incomplete gamma
function. 

Cheers,
	Paul

____________________________________________________________________ 
Paul Abbott                                   Phone: +61-8-9380-2734
Department of Physics                           Fax: +61-8-9380-1014
The University of Western Australia           
Nedlands WA  6907                     mailto:paul@physics.uwa.edu.au 
AUSTRALIA                        http://www.physics.uwa.edu.au/~paul

            God IS a weakly left-handed dice player
____________________________________________________________________