ASYMPTOTICS OF THE GENERALIZED EXPONENTIAL INTEGRAL, AND ERROR-BOUNDSIN THE UNIFORM ASYMPTOTIC SMOOTHING OF ITS STOKES DISCONTINUITIES

Uniform asymptotic approximations are derived for the generalized expo nential integral E(p)(z), where p is real and z complex. Both the case s p --> infinity and \z\ --> infinity are considered. For the case p - -> infinity an expansion in inverse powers of p is derived, which invo lves elementary functions and readily computed coefficients, and is un iformly valid for -pi + delta less than or equal to arg(z) less than o r equal to pi - delta (where delta is an arbitrary small positive cons tant). An approximation for large p involving the complementary error function is also derived, which is valid in an unbounded z-domain whic h contains the negative real axis. The case \z\ --> infinity is then c onsidered, and uniform asymptotic approximations are derived, which in volve 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 th an 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 initi ated by Berry. The novelty of the new Stokes smoothing approximations is that they include explicit and realistic error bounds, as do all th e other approximations in the present investigation.