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Asymptotics of the Generalized Exponential Integral, and Error Bounds in the Uniform Asymptotic Smoothing of Its Stokes Discontinuities

T. M. Dunster
Proceedings: Mathematical, Physical and Engineering Sciences
Vol. 452, No. 1949 (Jun. 8, 1996), pp. 1351-1367
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/52920
Page Count: 17
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Asymptotics of the Generalized Exponential Integral, and Error Bounds in the Uniform Asymptotic Smoothing of Its Stokes Discontinuities
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Abstract

Uniform asymptotic approximations are derived for the generalized exponential integral Ep(z), where p is real and z complex. Both the cases p → ∞ and |z|→ ∞ are considered. For the case p → ∞ an expansion in inverse powers of p is derived, which involves elementary functions and readily computed coefficients, and is uniformly valid for -π + δ ≤ arg(z) ≤ π - δ (where δ 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|→ ∞ 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 ≤ p ≤|z| + a, where a is bounded, uniformly for -π + δ ≤ arg(z) ≤ 3π - δ . 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.

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