# §6.4 Analytic Continuation

Analytic continuation of the principal value of $E_{1}\left(z\right)$ yields a multi-valued function with branch points at $z=0$ and $z=\infty$. The general value of $E_{1}\left(z\right)$ is given by

 6.4.1 $E_{1}\left(z\right)=\mathrm{Ein}\left(z\right)-\operatorname{Ln}z-\gamma;$

compare (6.2.4) and (4.2.6). Thus

 6.4.2 $E_{1}\left(ze^{2m\pi i}\right)=E_{1}\left(z\right)-2m\pi i,$ $m\in\mathbb{Z}$,

and

 6.4.3 $E_{1}\left(ze^{\pm\pi i}\right)=\mathrm{Ein}\left(-z\right)-\ln z-\gamma\mp\pi i,$ $|\operatorname{ph}z|\leq\pi$.

The general values of the other functions are defined in a similar manner, and

 6.4.4 $\displaystyle\mathrm{Ci}\left(ze^{\pm\pi i}\right)$ $\displaystyle=\pm\pi i+\mathrm{Ci}\left(z\right),$ 6.4.5 $\displaystyle\mathrm{Chi}\left(ze^{\pm\pi i}\right)$ $\displaystyle=\pm\pi i+\mathrm{Chi}\left(z\right),$ 6.4.6 $\displaystyle\mathrm{f}\left(ze^{\pm\pi i}\right)$ $\displaystyle=\pi e^{\mp iz}-\mathrm{f}\left(z\right),$ 6.4.7 $\displaystyle\mathrm{g}\left(ze^{\pm\pi i}\right)$ $\displaystyle=\mp\pi ie^{\mp iz}+\mathrm{g}\left(z\right).$

Unless indicated otherwise, in the rest of this chapter and elsewhere in the DLMF the functions $E_{1}\left(z\right)$, $\mathrm{Ci}\left(z\right)$, $\mathrm{Chi}\left(z\right)$, $\mathrm{f}\left(z\right)$, and $\mathrm{g}\left(z\right)$ assume their principal values, that is, the branches that are real on the positive real axis and two-valued on the negative real axis.