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Zbl 0581.33001
Temme, N.M.
Laplace type integrals: Transformation to standard form and uniform asymptotic expansions. (English)
[J] Q. Appl. Math. 43, 103-123 (1985). ISSN 0033-569X; ISSN 1552-4485/e

Laplace integrals of the form $$ F\sb{\lambda}(z)=(1/\Gamma (\lambda))\int\sp{\infty}\sb{0}t\sp{\lambda -1}e\sp{-zt}f(t)dt $$ are considered for large values of z; f is holomorphic in a domain that contains the non-negative reals. The ratio $\mu =\lambda /z$ is considered as a uniformity parameter in [0,$\infty)$. Integrals with the same asymptotic phenomenae are transformed into the above standard form by means of a canonical transformation. The analytic properties of this mapping are investigated, especially for the case that the mapping depends on $\mu$. Error bounds for the remainders in the asymptotic expansions are given. Applications include a ratio of gamma functions, modified Bessel functions and parabolic cylinder functions. Analogue results are considered for loop integrals in the complex plane. This is the second paper in a series of three; the first paper has been published in Analysis 3, 221-249 (1983; Zbl 0541.41036).

MSC 2000:: *33B15 Gamma-functions, etc.
33C10 Cylinder functions, etc.
41A60 Asymptotic problems in approximation
44A10 Laplace transform

Keywords: uniform asymptotic expansion; Error bounds; gamma functions; modified Bessel functions; parabolic cylinder functions

Citations: Zbl 0541.41036

Cited in: Zbl 0784.11007 Zbl 0641.33002

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