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Laplace type integrals: Transformation to standard form and uniform asymptotic expansions. (English) Zbl 0581.33001
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).

33B15Gamma, beta and polygamma functions
33C10Bessel and Airy functions, cylinder functions, ${}_0F_1$
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
44A10Laplace transform