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## Re: [Axiom-developer] Axiom and the Numerical Mathematics Consortium

 From: Doug Stewart Subject: Re: [Axiom-developer] Axiom and the Numerical Mathematics Consortium Date: Thu, 17 Jan 2008 11:40:14 -0500 User-agent: Thunderbird 1.5.0.14 (Windows/20071210)

address@hidden wrote:

I have been concentrating on Axiom's numerical capabilities. The last patch is the beginnings of regression tests against
Abramowitz and Stegun (1985) and Zwillinger's CRC Standard (2003).
I've also created firefox hyperdoc pages for the gamma function
standard from the new DLMF. I plan to fill these pages out with
Spad code and test cases as time permits.

I'm a member of the Numerical Mathematics Consortium
(http://www.nmconsortium.org).A recently published draft
standard, which I'm reviewing, is available at:
<http://www.nmconsortium.org/docs/NMC_Technical_Specification%20(9-24-2007).pdf>

The A&S handbook lists polynomial coefficients for approximation of E1,
the exponential integral. Does anyone know how these coefficients were
derived? Is it a chebyshev polynomial? I want to dynamically compute
these coefficients to the required precision.

Tim

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The exponential integral can be written as a special case of the incomplete gamma function <http://en.wikipedia.org/wiki/Incomplete_gamma_function>:

{\rm E}_n(x) =x^{n-1}\Gamma(1-n,x).\,

The exponential integral may also be generalized to

E_n(x) = \int_1^\infty \frac{e^{-xt}}{t^n}\, dt

this is from

http://en.wikipedia.org/wiki/Exponential_integral