Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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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

$\displaystyle {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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DOI: https://doi.org/10.1090/qam/782260
Article copyright: © Copyright 1985 American Mathematical Society


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