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Laplace method for asymptotic expansions of integrals

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1: 2.4 Contour Integrals
§2.4(iii) Laplace’s Method
2: 2.3 Integrals of a Real Variable
§2.3(iii) Laplace’s Method
3: 2.5 Mellin Transform Methods
§2.5 Mellin Transform Methods
§2.5(ii) Extensions
The Mellin transform method can also be extended to derive asymptotic expansions of multidimensional integrals having algebraic or logarithmic singularities, or both; see Wong (1989, Chapter 3), Paris and Kaminski (2001, Chapter 7), and McClure and Wong (1987). …
§2.5(iii) Laplace Transforms with Small Parameters
For examples in which the integral defining the Mellin transform h ( z ) does not exist for any value of z , see Wong (1989, Chapter 3), Bleistein and Handelsman (1975, Chapter 4), and Handelsman and Lew (1970).
4: Bibliography N
  • D. Naylor (1990) On an asymptotic expansion of the Kontorovich-Lebedev transform. Applicable Anal. 39 (4), pp. 249–263.
  • D. Naylor (1996) On an asymptotic expansion of the Kontorovich-Lebedev transform. Methods Appl. Anal. 3 (1), pp. 98–108.
  • G. Nemes and A. B. Olde Daalhuis (2016) Uniform asymptotic expansion for the incomplete beta function. SIGMA Symmetry Integrability Geom. Methods Appl. 12, pp. 101, 5 pages.
  • G. Nemes (2013a) An explicit formula for the coefficients in Laplace’s method. Constr. Approx. 38 (3), pp. 471–487.
  • G. Nemes (2013b) Error bounds and exponential improvement for Hermite’s asymptotic expansion for the gamma function. Appl. Anal. Discrete Math. 7 (1), pp. 161–179.
  • 5: 11.6 Asymptotic Expansions
    §11.6 Asymptotic Expansions
    §11.6(i) Large | z | , Fixed ν
    §11.6(ii) Large | ν | , Fixed z
    More fully, the series (11.2.1) and (11.2.2) can be regarded as generalized asymptotic expansions2.1(v)). …
    6: 2.11 Remainder Terms; Stokes Phenomenon
    The rest of this section is devoted to general methods for increasing this accuracy. … Then by application of Laplace’s method (§§2.4(iii) and 2.4(iv)), we have … A simple example is provided by Euler’s transformation (§3.9(ii)) applied to the asymptotic expansion for the exponential integral6.12(i)): …
    7: Bibliography S
  • A. Sharples (1971) Uniform asymptotic expansions of modified Mathieu functions. J. Reine Angew. Math. 247, pp. 1–17.
  • A. Sidi (2010) A simple approach to asymptotic expansions for Fourier integrals of singular functions. Appl. Math. Comput. 216 (11), pp. 3378–3385.
  • A. Sidi (2011) Asymptotic expansion of Mellin transforms in the complex plane. Int. J. Pure Appl. Math. 71 (3), pp. 465–480.
  • K. Soni (1980) Exact error terms in the asymptotic expansion of a class of integral transforms. I. Oscillatory kernels. SIAM J. Math. Anal. 11 (5), pp. 828–841.
  • W. F. Sun (1996) Uniform asymptotic expansions of Hermite polynomials. M. Phil. thesis, City University of Hong Kong.
  • 8: Bibliography T
  • N. M. Temme (1979b) The asymptotic expansion of the incomplete gamma functions. SIAM J. Math. Anal. 10 (4), pp. 757–766.
  • N. M. Temme (1985) Laplace type integrals: Transformation to standard form and uniform asymptotic expansions. Quart. Appl. Math. 43 (1), pp. 103–123.
  • N. M. Temme (1990b) Uniform asymptotic expansions of a class of integrals in terms of modified Bessel functions, with application to confluent hypergeometric functions. SIAM J. Math. Anal. 21 (1), pp. 241–261.
  • N. M. Temme (1995c) Uniform asymptotic expansions of integrals: A selection of problems. J. Comput. Appl. Math. 65 (1-3), pp. 395–417.
  • N. M. Temme (2015) Asymptotic Methods for Integrals. Series in Analysis, Vol. 6, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ.
  • 9: Bibliography H
  • 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.
  • C. J. Howls and A. B. Olde Daalhuis (1999) On the resurgence properties of the uniform asymptotic expansion of Bessel functions of large order. Proc. Roy. Soc. London Ser. A 455, pp. 3917–3930.
  • G. Hunter and M. Kuriyan (1976) Asymptotic expansions of Mathieu functions in wave mechanics. J. Comput. Phys. 21 (3), pp. 319–325.
  • 10: Bibliography B
  • E. A. Bender (1974) Asymptotic methods in enumeration. SIAM Rev. 16 (4), pp. 485–515.
  • B. C. Berndt and R. J. Evans (1984) Chapter 13 of Ramanujan’s second notebook: Integrals and asymptotic expansions. Expo. Math. 2 (4), pp. 289–347.
  • R. Bo and R. Wong (1994) Uniform asymptotic expansion of Charlier polynomials. Methods Appl. Anal. 1 (3), pp. 294–313.
  • W. G. C. Boyd (1995) Approximations for the late coefficients in asymptotic expansions arising in the method of steepest descents. Methods Appl. Anal. 2 (4), pp. 475–489.
  • J. Brüning (1984) On the asymptotic expansion of some integrals. Arch. Math. (Basel) 42 (3), pp. 253–259.