A Finite-Interval Uniqueness Theorem for Bilateral Laplace Transforms

Two or more bilateral Laplace transforms with a complex argument “s” may be equal in a finite vertical interval when, in fact, the transforms correspond to different functions. In this article, we prove that the existence of a bilateral Laplace transform in any finite horizontal interval uniquely determines the corresponding function. The result appears to be new as we could not find it in the literature. The novelty of the result is that the interval need not contain zero, the function need not be nonnegative and need not be integrable. The result has a potential to be useful in the context of fitting probability distributions to data using Laplace transforms or moment generating functions.