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

[Doug Stewart]

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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> http://lists.nongnu.org/mailman/listinfo/axiom-developer
>
>   


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