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Error analysis in a uniform asymptotic expansion for the generalised exponential integral. (English) Zbl 0870.33002
Summary: Uniform asymptotic expansions are derived for the generalised exponential integral E p (z), where both p and z are complex. These are derived by examining the differential equation satisfied by E p (z), an equation which possesses a double turning point at z/p=-1. The expansions, which involve the complementary error function, together approximate E p (z) as |p|, uniformly for all non-zero complex z satisfying 0arg(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 conductive to numerical evaluation.
MSC:
33B20Incomplete beta and gamma functions
34E20Asymptotic singular perturbations, turning point theory, WKB methods (ODE)
34E05Asymptotic expansions (ODE)
30E10Approximation in the complex domain
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)