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

Uniform asymptotic approximations are derived for the generalized exponential integral E(p)(z), where p is real and z complex. Both the cases p --> infinity and \z\ --> infinity are considered. For the case p --> infinity an expansion in inverse powers of p is derived, which involves elementary functions and readily computed coefficients, and is uniformly valid for -pi + delta less than or equal to arg(z) less than or equal to 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 \z\ --> infinity 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 less than or equal to p less than or equal to \z\ + a, where a is bounded, uniformly for -pi + delta less than or equal to arg(z) less than or equal to 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.