Uniform, exponentially improved, asymptotic expansions for the generalized exponential integral

@article{Olver1991UniformEI,
  title={Uniform, exponentially improved, asymptotic expansions for the generalized exponential integral},
  author={Frank W. J. Olver},
  journal={Siam Journal on Mathematical Analysis},
  year={1991},
  volume={22},
  pages={1475-1489}
}
  • F. Olver
  • Published 1 September 1991
  • Mathematics
  • Siam Journal on Mathematical Analysis
By allowing the number of terms in an asymptotic expansion to depend on the asymptotic variable, it is possible to obtain an error term that is exponentially small as the asymptotic variable tends to its limit. This procedure is called “exponential improvement.” It is shown how to improve exponentially the well-known Poincare expansions for the generalized exponential integral (or incomplete Gamma function) of large argument. New uniform expansions are derived in terms of elementary functions… 
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References

Uniform asymptotic smoothing of Stokes’s discontinuities
  • M. Berry
  • Mathematics
    Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
  • 1989
Across a Stokes line, where one exponential in an asymptotic expansion maximally dominates another, the multiplier of the small exponential changes rapidly. If the expansion is truncated near its…