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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 \(|p|\to\infty\), uniformly for all non-zero complex \(z\) satisfying \(0\leq \arg (z/p)\leq 2\pi\). 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:

33B20 Incomplete beta and gamma functions (error functions, probability integral, Fresnel integrals)
34E20 Singular perturbations, turning point theory, WKB methods for ordinary differential equations
34E05 Asymptotic expansions of solutions to ordinary differential equations
30E10 Approximation in the complex plane
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
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