Image:AsymptoticExpansionE1.png

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[edit] Summary

Description

Divergence of the asymptotic series for the w:Exponential integral

{\rm E}_1(z) = \int_1^\infty \frac{e^{-tz}}{t}\, {\rm d}t,\qquad \Re(z) \ge 0~~.

The finctions

E_1(z,N)= \frac{\exp(-z)}{z} \sum_{n=0}^{N-1} \frac{n!}{(-z)^n}

approximate E1(z) ar ~{\rm Re}(z) \gg 1. the relative error

\frac{ E_1(x,N)-E_1(x) }{E_1(x)}

is plotted versus ~x~ for 3 < x < 9 for
~N=1~ (red),
~N=2~ (green),
~N=3~ (yellow),
~N=4~ (blue),
~N=5~ (pink).</be> The larger is ~x~, the more terms in this expansion can be taken into account for approximation of the function (and the better is the resulting approximation).
Source

self-made
Date

2008
Author

Domitori
Permission
(Reusing this image)

Use it please. It is free. The first formula is smooth, you can expand it an do that you like. Enjoy!
Other versions not yet reported. Make your own!

[edit] Licensing

Public domain I, the copyright holder of this work, hereby release it into the public domain. This applies worldwide.

In case this is not legally possible:
I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.

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File history

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Date/TimeDimensionsUserComment
current11:30, 4 February 20081,110×1,110 (49 KB)Domitori ({{Information |Description=Divergence of the asymptotic series for the w:Exponential integral :<math>{\rm E}_1(z) = \int_1^\infty \frac{e^{-tz}}{t}\, dt,\qquad \Re(z) \ge 0~~.</math> The fincitons :<math> E_1(z,N)= \frac{\exp(-z)}{z} \sum_{n=0}^{N-1)
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