§8.19 Generalized Exponential Integral

For $p,z\in ℂ$

Most properties of $E_p\left(z\right)$ follow straightforwardly from those of $\mathrm{\Gamma }(a,z)$. For an extensive treatment of $E_1\left(z\right)$ see Chapter 6.

When the path of integration excludes the origin and does not cross the negative real axis (8.19.2) defines the principal value of $E_p\left(z\right)$, and unless indicated otherwise in the DLMF principal values are assumed.

In Figures 8.19.2–8.19.5, height corresponds to the absolute value of the function and color to the phase. See About Color Map.

For $n=1,2,3,\mathrm{\dots }$,

and

with $|\mathrm{ph}z|\le \pi$ in both equations. For $\psi \left(x\right)$ see §5.2(i).

When $p\in ℂ$

again with $|\mathrm{ph}z|\le \pi$ in both equations. The right-hand sides are replaced by their limiting forms when $p=1,2,3,\mathrm{\dots }$.

For $U(a,b,z)$ see §13.2(i).

See also Cuyt et al. (2008, pp. 277–285).

The general function $E_p\left(z\right)$ is attained by extending the path in (8.19.2) across the negative real axis. Unless $p$ is a nonpositive integer, $E_p\left(z\right)$ has a branch point at $z=0$. For $z\ne 0$ each branch of $E_p\left(z\right)$ is an entire function of $p$.

For $n=1,2,3,\mathrm{\dots }$ and $x>0$,

where

For the hypergeometric function $F(a,b;c;z)$ see §15.2(i). When $p=1,2,3,\mathrm{\dots }$, $L(p)$ can also be evaluated via (8.19.24).

For collections of integrals involving $E_p\left(z\right)$, especially for integer $p$, see Apelblat (1983, §§7.1–7.2) and LeCaine (1945).

For higher-order generalized exponential integrals see Meijer and Baken (1987) and Milgram (1985).