Laplace Transforms

Abstract

The function g(p) of the complex variable p defined by the integral

$$g(p) = \int\limits_0^\infty {f(t)e^{ - pt} dt}$$

is called the one sided Laplace transform of f(t) where f(t) is a function of the real variable t,(0 < t < ∞) which is integrable in every finite interval. If the integral converges at a point p = p₀, then it converges for every p such that Re p > Re p₀. The behavior of the integral (1) in the p-plane may be one of the following:

• (a) Divergent everywhere

• (b) Convergent everywhere

• (c) There exists a number 3 such that (1) converges, when Re p > β and diverges when Re p < β. The number β which is the greatest lower bound of Re p for the set of all p’s in the p-plane at which (1) converges is called the abscissa of convergence.