Asymptotics of the generalized exponential integral, and error bounds in the uniform asymptotic smoothing of its Stokes discontinuities.

*(English)* Zbl 0866.41025
Summary: Uniform asymptotic approximations are derived for the generalized exponential integral ${E}_{p}\left(z\right)$, where $p$ is real and $z$ complex. Both the cases $p\to \infty $ and $\left|z\right|\to \infty $ are considered. For the case $p\to \infty $ an expansion in inverse powers of $p$ is derived, which involves elementary functions and readily computed coefficients, and is uniformly valid for $-\pi +\delta \le arg\left(z\right)\le \pi -\delta $ (where $\delta $ is an arbitrary small positive constant). An approximation for large $p$ involving the complementary error function is also derived, which is valid in an unbounded $z$-domain which contains the negative real axis. The case $\left|z\right|\to \infty $ is then considered, and uniform asymptotic approximations are derived, which involve the complementary error function in the first approximation, and the parabolic cylinder function in an expansion. Both approximations are valid for values of $p$ satisfying $0\le p\le \left|z\right|+a$, where $a$ is bounded, uniformly for $-\pi +\delta \le arg\left(z\right)\le 3\pi -\delta $. These are examples of the so-called Stokes smoothing theory which was initiated by Berry. The novelty of the new Stokes smoothing approximations is that they include explicit and realistic error bounds, as do all the other approximations in the present investigation.

##### MSC:

41A60 | Asymptotic approximations, asymptotic expansions (steepest descent, etc.) |