Editing Exponential integral (section)

===Definition by Ein===

Both <math>\operatorname{Ei}</math> and <math>E_1</math> can be written more simply using the [[entire function]] <math>\operatorname{Ein}</math><ref>Abramowitz and Stegun, p.&nbsp;228, see footnote 3.</ref> defined as
:<math>
\operatorname{Ein}(z)
= \int_0^z (1-e^{-t})\frac{dt}{t}
= \sum_{k=1}^\infty \frac{(-1)^{k+1}z^k}{k\; k!}
</math>
(note that this is just the alternating series in the above definition of <math>E_1</math>). Then we have
:<math>
E_1(z) \,=\, -\gamma-\ln z + {\rm Ein}(z)
\qquad \left| \operatorname{Arg}(z) \right| < \pi
</math>
:<math>\operatorname{Ei}(x) \,=\, \gamma+\ln{x} - \operatorname{Ein}(-x)
\qquad x \neq 0
</math>
The function <math>\operatorname{Ein}</math> is related to the exponential generating function of the [[harmonic numbers]]:
:<math>
\operatorname{Ein}(z) = e^{-z} \, \sum_{n=1}^\infty \frac {z^n}{n!} H_n
</math>