Laplace Transformation: Further Topics

Abstract

Remember that if the Laplace integral

$$F\left( s \right): = \int_0^\infty {e^{ - st} } f\left( t \right)dt$$

converges for some finite real value s = x₀, it converges for all complex s with Rs > x₀ and thus defines a holomorphic or analytic function on the right-half plane Rs > x₀. The smallest of all such x₀’s is called the abscissa of simple convergence and the corresponding right half-plane is called the half-plane of simple convergence. Fortunately, the situation in most applications is more convenient with the Laplace integral converging absolutely for a finite value of s = x₁ ɛℝ, i.e.,

$$ \int_0^\infty {e^{ - x_1 t} \left| {f\left( t \right)} \right|} dt < 8. $$

It can then be shown that the Laplace integral absolutely converges for all s in the right half-plane Rs > x₁, which is called the half-plane of absolute convergence. The smallest of all such x₁’s is called the abscissa of absolute convergence. We always have x₀ ≤ x₁, i.e. the half-plane of absolute convergence is contained in the half-plane of simple convergence (absolute convergence implies simple convergence).