Laplace type integrals: transformation to standard form and uniform asymptotic expansions

Integrals are considered which can be transformed into the Laplace integral \[ F λ ( z ) = 1 Γ ( λ ) ∫ 0 ∞ t λ − 1 e − z t f ( t ) d t {F_\lambda } \left ( z \right ) = \frac {1}{{\Gamma \left ( \lambda \right )}}\int _0^\infty {{t^{\lambda - 1}}{e^{ - zt}}f\left ( t \right )dt} \] , where f f is holomorphic, z z is a large parameter, μ = λ / z \mu = \lambda /z is a uniformity parameter, μ ≥ 0 \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.