# Laplace method for asymptotic expansions of integrals

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##### 3: 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 $\mathscr{M}\mskip-3.0muh\mskip 3.0mu\left(z\right)$ 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: 11.11 Asymptotic Expansions of Anger–Weber Functions
11.11.10 $\mathbf{A}_{-\nu}\left(\lambda\nu\right)\sim-\frac{1}{\pi}\sum_{k=0}^{\infty}% \frac{(2k)!\,a_{k}(-\lambda)}{\nu^{2k+1}},$ $\nu\to\infty$, $|\operatorname{ph}\nu|\leq\pi-\delta$.
11.11.11 $\mathbf{A}_{-\nu}\left(\lambda\nu\right)\sim\left(\frac{2}{\pi\nu}\right)^{1/2% }e^{-\nu\mu}\sum_{k=0}^{\infty}\frac{{\left(\tfrac{1}{2}\right)_{k}}b_{k}(% \lambda)}{\nu^{k}},$ $\nu\to\infty$, $|\operatorname{ph}\nu|\leq\frac{\pi}{2}-\delta$,
##### 5: 11.6 Asymptotic Expansions
###### §11.6(ii) Large $|\nu|$, Fixed $z$
More fully, the series (11.2.1) and (11.2.2) can be regarded as generalized asymptotic expansions2.1(v)). …
##### 6: 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 (2020) An extension of Laplace’s method. Constr. Approx. 51 (2), pp. 247–272.
• ##### 7: 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)): …
##### 8: 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.
• ##### 9: Bibliography T
• 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.
• N. M. Temme (2022) Asymptotic expansions of Kummer hypergeometric functions for large values of the parameters. Integral Transforms Spec. Funct. 33 (1), pp. 16–31.
• ##### 10: 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.