Constructive Approximation

, Volume 18, Issue 3, pp 355–385 | Cite as

Exponential Asymptotics of the Mittag—Leffler Function

Article

Abstract.

The Stokes lines/curves are identified for the Mittag—Leffler function
$$ E_{\alpha, \beta}(z)=\sum^{\infty}_{n=0}\frac{z^n}{\Gamma(\alpha n+\beta)},\qquad\mathop{\rm Re}\nolimits \:\alpha > 0. $$
When α is not real, it is found that the Stokes curves are spirals. Away from the Stokes lines/curves, exponentially improved uniform asymptotic expansions are obtained. Near the Stokes lines/curves, Berry-type smooth transitions are achieved via the use of the complementary error function.
Key words. Mittag—Leffler function, Stokes lines/curves, Exponential asymptotics, Berry-type smooth transition. AMS Classification. Primary 41A60; Secondary 33E12. 

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Copyright information

© Springer-Verlag New York Inc. 2002

Authors and Affiliations

  • Wong
    • 1
  • Zhao
    • 2
  1. 1.Department of Mathematics City University of Hong Kong Tat Chee Avenue Kowloon Hong KongHK
  2. 2.Mathematics Department ZhongShan University GuangZhou 510275 P. R. ChinaCN

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