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Unformatted text preview: The Laplace Transform Shubhodeep Mukherji December 14, 2010 1 Definition, Existence, and Properties of the Laplace Transform 1.1 Definitions 1.1.1 The Laplace Transform Given a function F of the variable t defined for t > 0 and let s be a real variable. Consider the function f defined by f ( s ) = Z ∞ e st F ( t ) dt The function f defined by the integral above is called the Laplace transform of the function F . We denote the Laplace transform f of F by L { F ( t ) } . Let us now implement the definition of the Laplace Transform Example 1 Find the Laplace transform of the function F defined by F ( t ) = 1 Finding Laplace transforms using the definition is long, so mathematicians usually us a table of transforms. The last page of this handout is the Laplace transform. 1.1.2 Piecewise Continuity A function F is said to be piecewise continuous on a finite interval [ a,b ] if this interval can be divided into a finite number of subintervals such that F is continuous on the interior of each of these subintervals. 1.1.3 Exponential Order We say that F is of exponential order if a constant α exists such that the product e αt  F ( t )  is bounded for all sufficiently large values of t. Thus, if F is of exponential order and the values F ( t ) of F become infinite as t → ∞ , these values cannot become infinite faster than a multiple of M . 1 1.2 Existence of the Laplace Transform The Laplace Transform is defined by an improper integral. As such, we must consider its convergence in order to determine which types of fuctions possess Laplace transforms. Theorem 1 If F is piecewise continuous and of exponenetial order, then the Laplace transform Z ∞ e st F ( t ) dt of F exists....
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