## Abstract

The main applications of the Laplace transform are directed toward problems in which the time
where

*t*is the independent variable. We shall therefore use this variable in this chapter. Let*f*(*t*) be a complex-valued function of the real variable*t*such that*f*(*t*)*e*^{-ct}is abolutely integrable over 0 <*t*< ∞, where*c*is a real number. Then the Laplace transform of*f*(*t*)*, t*≥ 0, is defined as$$
\tilde f(s) = \mathcal{L}\left\{ {f(t)} \right\} = \int_0^\infty {f(t)e^{ - st} dt, Re s > c,}
$$

(1)

*s*= σ +*i*ω. The Laplace transform defined by (1) has the following basic properties.## Keywords

Laplace Transform Heaviside Function Main Application Residue Theorem Inhomogeneous Equation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media New York 2004