Laplace type integrals: transformation to standard form and uniform asymptotic expansions
Author:
N. M. Temme
Journal:
Quart. Appl. Math. 43 (1985), 103-123
MSC:
Primary 44A10; Secondary 41A60
DOI:
https://doi.org/10.1090/qam/782260
MathSciNet review:
782260
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Abstract: Integrals are considered which can be transformed into the Laplace integral \[ {F_\lambda } \left ( z \right ) = \frac {1}{{\Gamma \left ( \lambda \right )}}\int _0^\infty {{t^{\lambda - 1}}{e^{ - zt}}f\left ( t \right )dt} \], where $f$ is holomorphic, $z$ is a large parameter, $\mu = \lambda /z$ is a uniformity parameter, $\mu \ge 0$. A uniform asymptotic expansion is given with error bounds for the remainders. Applications are given for special functions, with a detailed analysis for a ratio of gamma functions. Further applications are mentioned for Bessel functions and parabolic cylinder functions. Analogue results are given for loop integrals in the complex plane.
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Chow, Sh.-N and J. K. Hale, (1982), Methods of bifurcation theory, Springer-Verlag, New York, Heidelberg, Berlin
Erdélyi, A. and M. Wyman, (1963), The asymptotic evaluation of certain integrals, Arch. Rational Mech. Anal. 14 217–260
Luke, Y. L., (1969), The special functions and their approximations, Academic Press, New York
Olver, F. W. J., (1974), Asymptotics and special functions, Academic Press, New York
Temme, N. M., (1983), Uniform asymptotic expansions of Laplace integrals, Analysis, 3, 221–249
Wong, R., (1980), Error bounds for asymptotic expansions of integrals, SIAM Review, 22, 401–435
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Article copyright:
© Copyright 1985
American Mathematical Society
