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Asymptotics, Superasymptotics, Hyperasymptotics...

  • Michael Berry
Part of the NATO ASI Series book series (NSSB, volume 284)

Abstract

My purpose is to describe several recent developments in our understanding of divergent series and the accurate calculation of the functions they represent. All the work has been1,2 or is being published3, so this will be an informal account, emphasising the new concepts and illustrating them with pictures.

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References

  1. 1.
    M.V. Berry, Uniform asymptotic smoothing of Stokes’s discontinuities, Proc. Roy. Soc. Lond, A422: 7–21(1989).ADSGoogle Scholar
  2. 2.
    M.V. Berry and C.J. Howls, Hyperasymptotics, Proc. Roy. Soc. Lona, A430: 653–668 (1990).MathSciNetADSCrossRefGoogle Scholar
  3. 3.
    M.V. Berry and C.J. Howls, Hyperasymptotics for integrals with saddles, submitted to Proc. Roy. Soc. Lond. (1991).Google Scholar
  4. 4.
    G.H. Hardy, “Divergent Series”, Clarendon Press, Oxford (1949).zbMATHGoogle Scholar
  5. 5.
    G.G. Stokes, On the numerical calculation of a class of definite integrals and infinite series, Trans. Comb. Phil. Soc. 9: 379–407 (1847).Google Scholar
  6. 6.
    B.G. Levi, A super time to renormalize, Physics Today, April 1988: 25.Google Scholar
  7. 7.
    Lord Kelvin, The scientific work of Sir George Stokes, Nature 67: 337–338 (1903).ADSGoogle Scholar
  8. 8.
    R.B. Dingle, “Asymptotic Expansions: their Derivation and Interpretation”, Academic Press, New York and London (1973).zbMATHGoogle Scholar
  9. 9.
    G.G. Stokes, On the discontinuity of arbitrary constants which appear in divergent developments, Trans. Camb. Phil. Soc. 10: 106–128 (1864).ADSGoogle Scholar
  10. 10.
    J.E. Littlewood, preface to ref.4.Google Scholar
  11. 11.
    N.G. de Bruijn, “Asymptotic methods in analysis”, North-Holland, Amsterdam (1958).zbMATHGoogle Scholar
  12. 12.
    M.V. Berry, Stokes’ phenomenon; smoothing a Victorian discontinuity, Publ. Math.of the Institut des Hautes Études scientifique, 68: 211 – 221 (1989).CrossRefGoogle Scholar
  13. 13.
    R. Balian, G. Parisi, and A. Voros, Discrepancies from asymptotic series and their relation to complex classical trajectories, Phys. Rev, Lett 41: 141–1144(1978).ADSGoogle Scholar
  14. 14.
    F.W.J. Olver, “Asymptotics and special functions” Academic Press, New York and London (1974).Google Scholar
  15. 15.
    M. Abramowitz and I.A. Stegun, 1964, “Handbook of mathematical functions”, National Bureau of Standards, Washington (1964).zbMATHGoogle Scholar
  16. 16.
    F.W.J. Olver, On Stokes’ phenomenon and converging factors, in “Proceedings of International Symposium on Asymptotic and Computational Analysis (Manitoba, Winnipeg 1989)”, R. Wong, ed., Marcel Dekker, New York (1990), pp329–355.Google Scholar
  17. 17.
    M.J. Rakovic, and E.A. Solov’ev, Higher orders of semiclassical expansion for the one-dimensional Schrödinger equation, Phys. Rev. A40: 6692–6694 (1989).ADSGoogle Scholar
  18. 18.
    W.G.C. Boyd, Stieltjes transforms and the Stokes phenomenon, Proc. Roy. Soc. Lond., A429: 227–246 (1990).ADSGoogle Scholar
  19. 19.
    D.S. Jones, Uniform asymptotic remainders, in “Proceedings of International Symposium on Asymptotic and Computational Analysis (Manitoba, Winnipeg 1989)” R Wong, ed., Marcel Dekker, New York (1990), pp241–264.Google Scholar
  20. 20.
    M.V. Berry, Waves near Stokes lines, Proc. Roy. Soc. Lond, A427: 265–280 (1990).ADSGoogle Scholar
  21. 21.
    M.V. Berry, Histories of adiabatic quantum transitions, Proc. Roy. Soc. Lond, A429: 61–72 (1990).ADSGoogle Scholar
  22. 22.
    R. Lim and M.V. Berry, Submitted to J.Phys.A. (1991).Google Scholar
  23. 23.
    M.V. Berry and C.J. Howls, Stokes surfaces of diffraction catastrophes with codimension three, Nonlinearity, 3:281–291 (1990).MathSciNetADSzbMATHCrossRefGoogle Scholar
  24. 24.
    J. Écalle, “Les fonctions résurgentes “(3 volumes) Publ. Math. Université de Paris-Sud (1981), and “Cinq applications des fonctions résurgentes “, Preprint 84T62, Orsay (1984).Google Scholar
  25. 25.
    A. Voros, The return of the quartic oscillator. The complex WKB method, Ann. Inst H. Poincaré, 39:211–338 (1983).MathSciNetzbMATHGoogle Scholar
  26. 26.
    P. Flagolet and A.M. Odlyzko, Singularity analysis of generating functions, SLAM J. Discrete Math., 3 (2): 216–240 (1990).CrossRefGoogle Scholar
  27. 27.
    P2C2E=processes too complicated to explain, see S. Rushdie, “Haroun and the sea of stories”, Granta, London (1990).Google Scholar
  28. 28.
    M.V. Berry, Some quantum-to-classical asymptotics, in “Chaos and quantum physics”,Les Houches Lecture Series 52, M J Giannoni and A Voros, eds. North-Holland, Amsterdam (1991).Google Scholar
  29. 29.
    M.V. Berry and J.P. Keating, A rule for quantizing chaos?, J. Phys. A., 23: 4839–4849 (1990).MathSciNetADSzbMATHCrossRefGoogle Scholar
  30. 30.
    R. Balian and C. Bloch, Solution of the Schrödinger equation in terms of classical paths, Ann. Phys. (N.Y.) 85: 514–545 (1974).MathSciNetADSzbMATHCrossRefGoogle Scholar
  31. 31.
    J. Knoll and R. Schaeffer, Semiclassical scattering theory with complex trajectories. 1. Elastic waves, Ann. Phys. (N.Y.), 97: 307 (1976).MathSciNetADSCrossRefGoogle Scholar
  32. 32.
    M.V. Berry, Infinitely many Stokes smoothings in the Gamma Function, submitted to Proc. Roy. Soc. Lond, (1990).Google Scholar

Copyright information

© Plenum Press, New York 1991

Authors and Affiliations

  • Michael Berry
    • 1
  1. 1.H.H. Wills Physics LaboratoryBristolUK

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