# §6.4 Analytic Continuation

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

 6.4.1 $\mathop{E_{1}\/}\nolimits\!\left(z\right)=\mathop{\mathrm{Ein}\/}\nolimits\!% \left(z\right)-\mathop{\mathrm{Ln}\/}\nolimits z-\gamma;$

compare (6.2.4) and (4.2.6). Thus

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

and

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

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

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

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