Asymptotic evaluation of an integral transform of entire exponential type functions exactly terminating to a polynomial

Abstract

We consider the asymptotic evaluation of an integral transform of an entire exponential function by performing an integration in the complex plane where the paths avoid the poles of the integrand residing at the origin. We shall see that by employing the property of exponential type functions, we obtain an exact terminating result provided that the condition $b \lambda>\tau$ is satisfied for some positive constant $b$, where $\tau$ is the type of the function and $\lambda$ is the asymptotic parameter. In particular, we find that the result terminates to a polynomial in $\lambda$ as $\lambda \to \infty$.