Incomplete Laplace Integrals: Uniform Asymptotic Expansion with Application to the Incomplete Beta Function

Abstract

The incomplete Laplace integral \[ \frac{1}{{\Gamma (\lambda )}}\int_\alpha ^\infty {t^{\lambda - 1} e^{ - zt} f(t)dt} \] is considered for large values of z. Both $\lambda $ and $\alpha $ are uniformity parameters in $[0,\infty )$. The basic approximant is an incomplete gamma function, that is, the above integral with $f = 1$. Also, a loop integral in the complex plane is considered with the same asymptotic features. The asymptotic expansions are furnished with error bounds for the remainders in the expansions. The results of the paper combine four kinds of asymptotic problems considered earlier. An application is given for the incomplete beta function. The present investigations are a continuation of earlier works of the author for the above integral with $\alpha = 0$. The new results are significantly based on the previous case.