Given an asymptotic of the $E_1(z)$ complex function (from Gradshteyn & Ryzhik p. xxxv):
$$E_1(z) \sim \frac{e^{-z}}{z}\left[1 -\frac{1}{z} +\frac{2}{z^2} - \ldots \right]$$
Where is this expansion valid ? G & R say: $| \arg(z)| < \frac{3\pi}{2}$, but actually I don't really know what that means. If it goes from $-\frac{3}{2}\pi$ to $\frac{3}{2}\pi$ it largely covers the whole complex plane. For instance $\arg(z)=2\pi$ would formally not fulfill the given restriction, but I can interpret it as $\arg(z)=0$ which fulfills it.
In particular the function $E_1(z)$ has a branch cut on the negative real axis, is this expansion along the branch cut valid ? All points on the negative real axis have $|\arg(z)|=\pi$ which is smaller than $\frac{3}{2}\pi$.
EDIT My key question is: Is the given asymptotic development valid for $arg(z)=+(\pi - \epsilon)$ as well as for $arg(z)=-(\pi - \epsilon)$ ? And if not, for which case of the two is it valid ?
I will give a detailed discussion of the asymptotics of $E_1(z)$. Our starting point is the integral representation $$ E_1 (z) = \frac{{{\rm e}^{ - z} }}{z}\int_0^{ + \infty } {\frac{{{\rm e}^{ - t} }}{{1 + t/z}}{\rm d}t} , \qquad |\arg z|<\pi $$ (cf. $(6.2.2)$). For any non-negative integer $N$, we have $$ \frac{1}{{1 + t/z}} = \sum\limits_{n = 0}^{N - 1} {( - 1)^n \frac{1}{{z^n }}t^n } + ( - 1)^N \frac{1}{{z^N }}\frac{{t^N }}{{1 + t/z}} $$ provided $t\ge 0$ and $|\arg z|<\pi$. Substituting this into the integral representation for $E_1(z)$ and integrating term-by-term, we obtain $$ E_1 (z) = \frac{{{\rm e}^{ - z} }}{z}\left( {\sum\limits_{n = 0}^{N - 1} {( - 1)^n \frac{{n!}}{{z^n }}} + R_N (z)} \right), \tag{1} $$ where the remainder term $R_N(z)$ is given by $$ R_N (z) = ( - 1)^N \frac{1}{{z^N }}\int_0^{ + \infty } {\frac{{t^N {\rm e}^{ - t} }}{{1 + t/z}}{\rm d}t} , \qquad |\arg z|<\pi. $$ With the notation of this paper, $$ R_N (z) = ( - 1)^N \frac{{N!}}{{z^N }}\Lambda _{N + 1} (z),\qquad \Lambda _p (z) = \frac{1}{{\Gamma (p)}}\int_0^{ + \infty } {\frac{{t^{p - 1} {\rm e}^{ - t} }}{{1 + t/z}}{\rm d}t} , $$ where $\Lambda_p(z)$ may be expressed in terms of the incomplete gamma function: $\Lambda _p (z) = z^p {\rm e}^z \Gamma (1 - p,z)$. Thus $\Lambda_p(z)$ extends analytically to the Riemann surface associated with the logarithm. Therefore $(1)$ is valid for all values of $\arg z$ with $$ R_N (z) = ( - 1)^N \frac{{N!}}{{z^N }}\Lambda _{N + 1} (z), \quad \Lambda _p (z) \overset{\text{def}}{=} z^p {\rm e}^z \Gamma (1 - p,z). $$ By the results in Appendix B of the above cited paper, we deduce $$ \left| {R_N (z)} \right| \le \frac{{N!}}{{\left| z \right|^N }} \times \begin{cases} 1 & \text{if }\; |\arg z|\leq \cfrac{\pi}{2},\\[1ex] \min\left(|\csc(\arg z)|,\chi(N+1)+1\right) & \text{if }\; \cfrac{\pi}{2} <|\arg z|\leq \pi,\\[1ex] \cfrac{{\sqrt {2\pi (N + 1)} }}{{\left| {\cos (\arg z)} \right|^{N + 1} }} + \chi (N + 1) + 1 & \text{if }\; \pi <|\arg z|<\cfrac{3\pi}{2}, \end{cases}\tag{2} $$ with $$ \chi (p) \overset{\text{def}}{=} \sqrt \pi \frac{{\Gamma \left( {\frac{p}{2} + 1} \right)}}{{\Gamma \left( {\frac{p}{2} + \frac{1}{2}} \right)}} \sim \sqrt {\frac{\pi }{2}\left( {p + \frac{1}{2}} \right)} . $$ This bound shows that $R_N(z)=\mathcal{O}(z^{-N})$ as long as $|\arg z|\le \frac{3\pi}{2}-\delta<\frac{3\pi}{2}$, with any fixed small $\delta>0$.
The bound $(2)$ also shows that the asymptotic expansion becomes worse as $|\arg z|$ increases beyond $\pi$. However, it is enough to approximate $E_1(z)$ on $|\arg z|\le \pi$ and its value on other Riemann sheets follows from the connection formula $E_1 (z) = E_1 (z{\rm e}^{ \pm 2\pi {\rm i}m} ) \pm 2\pi {\rm i}m$ where $m=0,1,2,\ldots$. Thus for example $$\tag{3} E_1 (z) \sim \frac{{{\rm e}^{ - z} }}{z}\sum\limits_{n = 0}^\infty {( - 1)^n \frac{{n!}}{{z^n }}} $$ as $z\to \infty$ in the sector $\left| {\arg z} \right| \le \frac{{3\pi }}{2} - \delta$, and $$\tag{4} E_1 (z) \sim - 2\pi {\rm i} + \frac{{{\rm e}^{ - z} }}{z}\sum\limits_{n = 0}^\infty {( - 1)^n \frac{{n!}}{{z^n }}} $$ as $z\to \infty$ in the sector $ - \frac{\pi }{2} + \delta \le \arg z \le \frac{{7\pi }}{2} - \delta$. There seems to be a contradiction, for there are two different asymptotic expansions for $E_1(z)$ in the common sector of validity $ - \frac{\pi }{2} + \delta \le \arg z \le \frac{{3\pi }}{2} - \delta$. Note, however, that in this sector, any term of the (formal) expansion $$ \frac{{{\rm e}^{ - z} }}{z}\sum\limits_{n = 0}^\infty {( - 1)^n \frac{{n!}}{{z^n }}} $$ is exponentially larger than the term $- 2\pi {\rm i}$. Hence this constant term can be absorbed into the error term. As we mentioned above, the error bound $(2)$ suggests that $(3)$ should be used in the sector $|\arg z|\le \pi$ and $(4)$ should be used when $\pi \le \arg z \le 3\pi$. The abrupt change in the asymptotic form on the ray $\arg z=\pi$ is called a Stokes phenomenon and the ray $\arg z=\pi$ is called a Stokes line. Write $$ E_1 (z) = \frac{{{\rm e}^{ - z} }}{z}\sum\limits_{n = 0}^{N - 1} {( - 1)^n \frac{{n!}}{{z^n }}} + \frac{{{\rm e}^{ - z} }}{z}R_N (z). $$ From the numerical point of view, it makes sense to truncate the series at its numerically least term. Now it is not difficult to show that this occurs when $N\approx |z|$. Stirling's formula and the error bound $(2)$ shows that $$ \left| {\frac{{{\rm e}^{ - z} }}{z}R_N (z)} \right| \le \mathcal{O}(1){\rm e}^{ - z - \left| z \right|} ,\qquad N\approx |z|, $$ in a small neighbourhood of the Stokes line $\arg z=\pi$. In particular, when $\arg z =\pi$, ${\rm e}^{ - z - \left| z \right|} = 1$. Thus at optimal truncation, when $N\approx |z|$, the error term is of order $1$ on the Stokes line $\arg z=\pi$. This is when the inclusion of $-2\pi \mathrm{i}$ starts to make sense from a numerical point of view. More can be said but I stop here.