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# 6.1we - 6.1 THE LAPLACE TRANSFORM KIAM HEONG KWA A tool of...

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6.1: THE LAPLACE TRANSFORM KIAM HEONG KWA A tool of frequent occurrence in modern mathematics on studying an unknown function f is to transform the function f into another function by means of a linear integral transformation T : (1) f ( t ) 7→ T f ( s ) = T [ f ( t )]( s ) = Z b a K ( s, t ) f ( t ) dt. Equation (1) gives the value at the point s of the transformed function T f , called the transform of f , in terms of the values of f at points in the interval ( a, b ). Either a or b or both may be infinite. The function K of two variables is referred to as the kernel of the transformation T . Such a linear transformation T satisfies the linearity property (2) T ( αf + βg ) = α T f + β T g for all functions f and g under consideration and for all scalars α and β . The transformation T will be particularly useful if, from the knowl- edge of T f , we are able to determine f uniquely. The process of recov- ering the unknown function f from its transform T f is known as the inversion problem . In this case, it is customary to denote the inverse transformation by T - 1 , so that T - 1 ( T f ) = f. The usefulness of an integral transformation arises from the fact that manipulating the transformed functions is often equivalent to handling the unknown functions under consideration and may be simpler. In this way, an entire problem may be converted to a simpler one and solved and then the original problem is solved by inverting the transformation. We shall consider a particular type of linear integral transformation, called the one-sided Laplace transform : (3) f ( t ) 7→ L f ( s ) = L [ f ( t )]( s ) = Z 0 e - st f ( t ) dt = lim τ →∞ Z τ 0 e - st f ( t ) dt. Date : February 21, 2011. 1

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2 KIAM HEONG KWA The Laplace transform can also be defined for complex-valued func- tions with the parameter s being complex also[1, 2]. As indicated in the following example, not every function possesses a Laplace transform. Example 1. The Laplace transform L [ e t 2 ]( s ) is not well-defined for any s . This is so since for every s , if t * > max { 1 - | s | , 1 + | s |} , then Z τ t * e - st e t 2 dt = Z τ t * e t ( t - s ) dt Z τ t * e t dt = e τ - e t * for all τ t * , from which it follows that Z t * e - st e t 2 dt lim τ →∞ ( e τ - e t * ) = . This implies that the improper integral defining L [ e t 2 ]( s ) also diverges. In order to show that there is a large class of functions that admits Laplace transforms, we first make a few definitions in the following. Definition 1. A function f is said to have a jump discontinuity at a point τ if both the one-sided limits f ( τ - ) = lim t τ - f ( t ) and f ( τ +) = lim t τ + f ( t ) exist as finite numbers and f ( τ - ) 6 = f ( τ +) . Definition 2. A function f is said to be piecewise continuous on the interval [0 , ) if the following conditions hold: (1) f (0+) = lim t 0+ f ( t ) exists as a finite number, (2) f is continuous on every finite interval (0 , b ) except possibly at a finite number of points t 1 , t 2 , · · · , t n in (0 , b ) at which f has jump discontinuities.
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6.1we - 6.1 THE LAPLACE TRANSFORM KIAM HEONG KWA A tool of...

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