Error analysis in a uniform asymptotic expansion for the generalised exponential integral

Uniform asymptotic expansions are derived for the generalised exponential integral Ep(z), where both p and z are complex. These are derived by examining the differential equation satisfied by Ep(z), an equation which possesses a double turning point at z/p = −1. The expansions, which involve the complementary error function, together approximate Ep(z) as ¦p¦ → ∞, uniformly for all non-zero complex z satisfying 0 ⩽ arg(z/p) ⩽ 2π. The error terms associated with the truncated expansions are shown to be solutions of inhomogeneous differential equations, and from these explicit and realistic bounds are derived. By employing the Maximum-Modulus Theorem the bounds are then simplified to make them more conducive to numerical evaluation.