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

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
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 !
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 4

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online