Laplace Transformation: Further Topics

Urs Graf ()
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Urs Graf: Berne University of Applied Sciences, Biel School of Engineering and Architecture

Chapter Chapter 6 in Applied Laplace Transforms and z-Transforms for Scientists and Engineers, 2004, pp 215-286 from Springer

Abstract: 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 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).

Keywords: Asymptotic Expansion; Branch Point; Image Function; Laplace Transformation; Integration Contour (search for similar items in EconPapers)
Date: 2004
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DOI: 10.1007/978-3-0348-7846-3_6

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