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Mittag-Leffler expansion

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1: 1.10 Functions of a Complex Variable
Mittag-Leffler’s Expansion
2: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
§10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
For exponentially-improved asymptotic expansions in the same circumstances, together with smooth interpretations of the corresponding Stokes phenomenon (§§2.11(iii)2.11(v)) see Wong and Zhao (1999b) when ρ > 0 , and Wong and Zhao (1999a) when - 1 < ρ < 0 . The Laplace transform of ϕ ( ρ , β ; z ) can be expressed in terms of the Mittag-Leffler function: …This reference includes exponentially-improved asymptotic expansions for E a , b ( z ) when | z | , together with a smooth interpretation of Stokes phenomena. See also Wong and Zhao (2002a), and for further information on the Mittag-Leffler function see Erdélyi et al. (1955, §18.1), Paris and Kaminski (2001, §5.1.4), and Haubold et al. (2011). …
3: Bibliography P
  • R. B. Paris (2001a) On the use of Hadamard expansions in hyperasymptotic evaluation. I. Real variables. Proc. Roy. Soc. London Ser. A 457 (2016), pp. 2835–2853.
  • R. B. Paris (2001b) On the use of Hadamard expansions in hyperasymptotic evaluation. II. Complex variables. Proc. Roy. Soc. London Ser. A 457, pp. 2855–2869.
  • R. B. Paris (2002a) Error bounds for the uniform asymptotic expansion of the incomplete gamma function. J. Comput. Appl. Math. 147 (1), pp. 215–231.
  • R. B. Paris (2002b) A uniform asymptotic expansion for the incomplete gamma function. J. Comput. Appl. Math. 148 (2), pp. 323–339.
  • R. B. Paris (2002c) Exponential asymptotics of the Mittag-Leffler function. Proc. Roy. Soc. London Ser. A 458, pp. 3041–3052.
  • 4: Bibliography W
  • R. Wong and Y. Zhao (2002a) Exponential asymptotics of the Mittag-Leffler function. Constr. Approx. 18 (3), pp. 355–385.
  • R. Wong (1973b) On uniform asymptotic expansion of definite integrals. J. Approximation Theory 7 (1), pp. 76–86.
  • R. Wong (1981) Asymptotic expansions of the Kontorovich-Lebedev transform. Appl. Anal. 12 (3), pp. 161–172.
  • R. Wong (1983) Applications of some recent results in asymptotic expansions. Congr. Numer. 37, pp. 145–182.
  • E. M. Wright (1935) The asymptotic expansion of the generalized Bessel function. Proc. London Math. Soc. (2) 38, pp. 257–270.
  • 5: Bibliography H
  • P. I. Hadži (1976a) Expansions for the probability function in series of Čebyšev polynomials and Bessel functions. Bul. Akad. Štiince RSS Moldoven. 1976 (1), pp. 77–80, 96 (Russian).
  • R. A. Handelsman and J. S. Lew (1970) Asymptotic expansion of Laplace transforms near the origin. SIAM J. Math. Anal. 1 (1), pp. 118–130.
  • R. A. Handelsman and J. S. Lew (1971) Asymptotic expansion of a class of integral transforms with algebraically dominated kernels. J. Math. Anal. Appl. 35 (2), pp. 405–433.
  • F. E. Harris (2000) Spherical Bessel expansions of sine, cosine, and exponential integrals. Appl. Numer. Math. 34 (1), pp. 95–98.
  • H. J. Haubold, A. M. Mathai, and R. K. Saxena (2011) Mittag-Leffler functions and their applications. J. Appl. Math. 2011, pp. Art. ID 298628, 51 pages.