Asymptotic expansion of some Laplace integrals with explicit error terms

The asymptotic expansion of the Laplace integral,

I(z) = ∫₀∞e^−ztf(t) dt, |z| →∞ , |ang z| < π/2, where f(t) is a function whose behavior near the origin is known, is generally obtained by Watson's Lemma.

In many applications, f(t) is a function analytic in the right half complex t-plane except that it may have a branch point at the origin. There are many completely different and comparatively more sophisticated techniques being used to obtain the asymptotic expansion of I(z) as |z| → ∞z. Dr. K. Soni's technique for obtaining a uniform asymptotic expansion for finite Laplace transform is comparatively simple and less restrictive. Furthermore, it provides the remainder term explicitly when the expansion is terminated after a finite number of terms. We use this technique to obtain the asymptotic expansion of I(z) together with an explicit representation for the error term. By considering different examples we show how the error, involved in replacing I(z) by a finite number of terms in the asymptotic expansion, can be estimated.