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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 λ (z)=(1/Γ(λ)) 0 t λ-1 e -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 μ=λ/z is considered as a uniformity parameter in [0,). 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 μ. 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:
33B15Gamma, beta and polygamma functions
33C10Bessel and Airy functions, cylinder functions, 0 F 1
41A60Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
44A10Laplace transform