Ch 6.1: Definition of Laplace TransformThe Laplace TransformExample 1Example 2Piecewise Continuous FunctionsExample 3Example 4Theorem 6.1.2Example 5Example 6Example 7Linearity of the Laplace TransformExample 8Ch 6.1: Definition of Laplace TransformMany practical engineering problems involve mechanical or electrical systems acted upon by discontinuous or impulsive forcing terms. For such problems the methods described in Chapter 3 are difficult to apply. In this chapter we use the Laplace transform to convert a problem for an unknown function f into a simpler problem for F, solve for F, and then recover f from its transform F. Given a known function K(s,t), an integral transform of a function f is a relation of the form ,)(),()( dttftsKsFThe Laplace TransformLet f be a function defined for t  0, and satisfies certain conditions to be named later. The Laplace Transform of f is defined asThus the kernel function is K(s,t) = e-st. Since solutions of linear differential equations with constant coefficients are based on the exponential function, the Laplace transform is particularly useful for such equations. Note that the Laplace Transform is defined by an improper integral, and thus must be checked for convergence. On the next few slides, we review examples of improper integrals and piecewise continuous functions.  0)()()( dttfesFtfLstExample 1Consider the following improper integral.We can evaluate this integral as follows:Note that if s = 0, then est = 1. Thus the following two cases hold:  1lim1limlim000sbbbstbbstbstessedtedte0dtest0. if,divergesand0; if,100sdtessdteststExample 2Consider the following improper integral.We can evaluate this integral using integration by parts:Since this limit diverges, so does the original integral.   1cossinlimcossinlimsinsinlimcoslimcos000000bsbsbtststtdtststtdtsttdtstbbbbbbbbb0costdtstPiecewise Continuous FunctionsA function f is piecewise continuous on an interval [a, b] if this interval can be partitioned by a finite number of pointsa = t0 < t1 < … < tn = b such that (1) f is continuous on each (tk, tk+1) In other words, f is piecewise continuous on [a, b] if it is continuous there except for a finite number of jump discontinuities.nktfnktfkktttt,,1,)(lim)3(1,,0,)(lim)2(1Example 3Consider the following piecewise-defined function f.From this definition of f, and from the graph of f below, we see that f is piecewise continuous on [0, 3].32121,310,)(2tttttttfExample 4Consider the following piecewise-defined function f.From this definition of f, and from the graph of f below, we see that f is not piecewise continuous on [0, 3]. 32,421,210,1)(12ttttttfTheorem 6.1.2Suppose that f is a function for which the following hold:(1) f is piecewise continuous on [0, b] for all b > 0. (2) | f(t) |  Keat when t  M, for constants a, K, M, with K, M > 0.Then the Laplace Transform of f exists for s > a.Note: A function f that satisfies the conditions specified above is said to to have exponential order as t  . finite )()()(0 dttfesFtfLstExample 5Let f (t) = 1 for t  0. Then the Laplace transform F(s) of f is: 0,1limlim1000sssedtedteLbstbbstbstExample 6Let f (t) = eat for t  0. Then the Laplace transform F(s) of f is: asasasedtedteeeLbtasbbtasbatstat,1limlim0)(0)(0Example 7Let f (t) = sin(at) for t  0. Using integration by parts twice, the Laplace transform F(s) of f is found as follows: 0,)()(1sin/)sin(lim1coslim1cos/)cos(limsinlimsin)sin()(22220000000sasasFsFasaateasaateasaateasaateasaateatdteatdteatLsFbstbstbbstbbstbstbbstbstLinearity of the Laplace TransformSuppose f and g are functions whose Laplace transforms exist for s > a1 and s > a2, respectively.Then, for s greater than the maximum of a1 and a2, the Laplace transform of c1 f (t) + c2g(t) exists. That is,with  finite is )()()()(02121 dttgctfcetgctfcLst    )( )( )( )()()(21020121tgLctfLcdttgecdttfectgctfcLststExample 8Let f (t) = 5e-2t - 3sin(4t) for t  0. Then by linearity of the Laplace transform, and using results of previous examples, the Laplace transform F(s) of f is:   