Constructive Approximation

, Volume 38, Issue 3, pp 471–487 | Cite as

An Explicit Formula for the Coefficients in Laplace’s Method

Article

Abstract

Laplace’s method is one of the fundamental techniques in the asymptotic approximation of integrals. The coefficients appearing in the resulting asymptotic expansion arise as the coefficients of a convergent or asymptotic series of a function defined in an implicit form. Due to the tedious computation of these coefficients, most standard textbooks on asymptotic approximations of integrals do not give explicit formulas for them. Nevertheless, we can find some more or less explicit representations for the coefficients in the literature: Perron’s formula gives them in terms of derivatives of an explicit function; Campbell, Fröman and Walles simplified Perron’s method by computing these derivatives using an explicit recurrence relation. The most recent contribution is due to Wojdylo, who rediscovered the Campbell, Fröman and Walles formula and rewrote it in terms of partial ordinary Bell polynomials. In this paper, we provide an alternative representation for the coefficients that contains ordinary potential polynomials. The proof is based on Perron’s formula and a theorem of Comtet. The asymptotic expansions of the gamma function and the incomplete gamma function are given as illustrations.

Keywords

Asymptotic expansions Laplace’s method Partial Bell polynomials Potential polynomials Perron’s formula 

Mathematics Subject Classification (2010)

41A60 41A58 

References

  1. 1.
    Brassesco, S., Méndez, M.A.: The asymptotic expansion for n! and the Lagrange inversion formula. Ramanujan J. 24, 219–234 (2011) MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Campbell, J.A., Fröman, P.O., Walles, E.: Explicit series formulae for the evaluation of integrals by the method of steepest descent. Stud. Appl. Math. 77, 151–172 (1987) MathSciNetMATHGoogle Scholar
  3. 3.
    Comtet, L.: Advanced Combinatorics. Reidel, Dordrecht (1974) CrossRefMATHGoogle Scholar
  4. 4.
    De Angelis, V.: Asymptotic expansions and positivity of coefficients for large powers of analytic functions. Int. J. Math. Math. Sci. 2003, 1003–1025 (2003) CrossRefMATHGoogle Scholar
  5. 5.
    Dunster, T.M., Paris, R.B., Cang, S.: On the high-order coefficients in the uniform asymptotic expansion for the incomplete gamma function. Methods Appl. Anal. 5, 223–247 (1998) MathSciNetMATHGoogle Scholar
  6. 6.
    Erdélyi, A.: Asymptotic Expansions. Dover, New York (1956) MATHGoogle Scholar
  7. 7.
    López, J.L., Pagola, P., Pérez Sinusía, E.: A simplification of Laplace’s method: applications to the Gamma function and Gauss hypergeometric function. J. Approx. Theory 161, 280–291 (2009) MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    López, J.L., Pagola, P.: An explicit formula for the coefficients of the saddle point method. Constr. Approx. 33, 145–162 (2011) MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Nemes, G.: On the coefficients of the asymptotic expansion of n!. J. Integer Seq. 13, 10.6.6 (2010) MathSciNetGoogle Scholar
  10. 10.
    Olver, F.W.J.: Error bounds for stationary phase approximations. SIAM J. Math. Anal. 5, 19–29 (1974) MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Olver, F.W.J.: Asymptotics and Special Functions. Academic Press, New York (1974). Reprinted, A.K. Peters, Wellesley (1997) Google Scholar
  12. 12.
    Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST Handbook of Mathematical Functions. Cambridge University Press, New York (2010) MATHGoogle Scholar
  13. 13.
    Paris, R.B., Kaminski, D.: Asymptotics and Mellin–Barnes Integrals. Cambridge University Press, Cambridge (2001) CrossRefMATHGoogle Scholar
  14. 14.
    Paris, R.B.: Hadamard Expansions and Hyperasymptotic Evaluation: An Extension of the Method of Steepest Descents. Cambridge University Press, Cambridge (2011) CrossRefGoogle Scholar
  15. 15.
    Perron, O.: Über die näherungsweise Berechnung von Funktionen großer Zahlen. Sitzungsber. Bayr. Akad. Wissensch. (Münch. Ber.), 191–219 (1917) Google Scholar
  16. 16.
    Riordan, J.: Combinatorial Identities. Wiley, New York (1979) Google Scholar
  17. 17.
    Temme, N.M.: Uniform asymptotic expansions of the incomplete gamma functions and the incomplete beta function. Math. Comput. 29, 1109–1114 (1975) MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Temme, N.M.: The asymptotic expansion of the incomplete gamma functions. SIAM J. Math. Anal. 10, 757–766 (1979) MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Wojdylo, J.: On the coefficients that arise from Laplace’s method. J. Comput. Appl. Math. 196, 241–266 (2006) MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Wojdylo, J.: Computing the coefficients in Laplace’s method. SIAM Rev. 48, 76–96 (2006) MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Wong, R.: Asymptotic Approximations of Integrals. Academic Press, New York (1989). Reprinted, Classics Appl. Math., vol. 34. SIAM, Philadelphia (2001) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mathematics and its ApplicationsCentral European UniversityBudapestHungary

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