# L72 - Fall 2003 Math 308/501502 7 Laplace Transforms 7.2...

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Fall 2003 Math 308/501–502 7 Laplace Transforms 7.2 Deﬁnition of the Laplace Transform Wed, 08/Oct c ± 2003, Art Belmonte Summary Deﬁnitions Given a function f of t ,the Laplace transform of f is a function L { f } ( s ) = F ( s ) deﬁned by L { f } ( s ) = ± 0 f ( t ) e - st dt = lim T →∞ ± T 0 f ( t ) e - , when this limit exists. Here s > 0 and may be further restricted depending on f . The transform is an operator which acts on a function to produce yet another function. A function f is piecewise continuous on ( 0 , ) if it has only ﬁnitely many jump discontinuities on any ﬁnite subinterval of ( 0 , ) . In a similar manner, f is piecewise differentiable on ( 0 , ) if it is continuous and its derivative is piecewise continuous. A function f is of exponential order if | f ( t ) | ≤ Ce at for t > 0, where C and a are constants. Existence Theorem for Laplace Transforms If f is a function deﬁned on [0 , ) that is piecewise continuous and of exponential order (say | f ( t ) | ≤ ), then the Laplace transform L { f } ( s ) exists for s > a (at least). Linearity of the Laplace Transform Let the Laplace transforms of f , f 1 ,and f 2 exist for s and let c be any constant. Then we have L { f 1 + f 2 } = L { f 1 } + L { f 2 } L { cf } = c L { f } Notes When computing Laplace transforms strictly with a pencil, we use integration by parts, L’Hospital’s rule, etc. On the other hand, the MATLAB Symbolic Math Toolbox (SMT) command laplace computes Laplace transforms at one fell swoop. In between these two extremes is a middle ground, where one mimicks hand work semiautomatically with the SMT commands int , subs limit . This is how Hand Examples below were computed. They were veriﬁed via laplace . Small Table of Laplace Transforms f ( t ) F ( s ) = L { f } ( s ) Restrictions 1 1 s s > 0 e 1 s - a s > a t n n !

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## This note was uploaded on 03/29/2008 for the course MATH 308 taught by Professor Comech during the Spring '08 term at Texas A&M.

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L72 - Fall 2003 Math 308/501502 7 Laplace Transforms 7.2...

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