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

Laplace integrals of the form

${F}_{\lambda }\left(z\right)=\left(1/{\Gamma }\left(\lambda \right)\right){\int }_{0}^{\infty }{t}^{\lambda -1}{e}^{-zt}f\left(t\right)dt$

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,$\infty \right)$. 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:
 33B15 Gamma, beta and polygamma functions 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$ 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.) 44A10 Laplace transform