Class Note for MATH 322 at UA
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Date Created: 02/06/15
AppI cat 1 Definitions Chapter 6 Laplace Transforms a The Laplace transform f of a piecewise continuous function f defined on 0 00 is given by fs Fs 0ltgt0 expis t ft dt 0 Clearly the above integral only converges if f does not grow too fast at infinity More precisely if there exist constants M gt 0 and k ER such that lffl S MeXPltf for 139 large enough then the Laplace transform of f exists for all s gt k o If f has a Laplace transform F we also say that f is the inverse Laplace transform of F and write f E 1F Chapter 6 Laplace Transforms Appu 2 Properties of the Laplace transform a The Laplace transform is a linear transformation ie if f1 and f2 have Laplace transforms and if 041 and 042 are constants then Ot1f1 042 041 f1 a2 f2 0 As for Fourier transforms the statement f 5 1 CW should be understood in a point wise fashion only at points where f is continuous 9 Since there is no explicit formula for the inverse Laplace transform formal inversion is accomplished by using tables shifting t and 5 taking derivatives of known Laplace transforms or integrating them Chapter 6 Laplace Transforms s shifting Laplace transform of derivatives 84 antiderivatives 9 Note All of the formulas written in what follows implicitly assume that the various functions used have well defined Laplace transforms One should therefore check that the corresponding Laplace transforms exist before using these formulas o s shifting formulas E eatft s Fsia a Laplace transform of derivatives a F s swxs 7 Ho z W s 52 fs e 5f0 e 0 eatrm 1Fs e a 139 Chapter 6 Laplace Translbrms Applluallulh39 of Laplace translm39ms Laplace transform of derivatives and antiderivatives o More generally a M 5 s fs7s 1f075 2f07 7f 10 o Laplace transform of antiderivatives Otf7 d7 5 fs Ot f7 d7 r1 Law 139 0 Examples 9 Find the Laplace transforms of sinwt and coswt 9 Find the inverse Laplace transforms of 1ss2 1 and 152s2 Chapter 6 Laplace Transharms o The Heaviside function or step function Ht is defined as 7 0 iftlt0 HtT1ift20 0 We can calculate that for a gt 0 E Ht7 a s e725 o More generally we have the following time shifting formulas for a gt 0 E ft 7 a Ht 7 a s e aS fs ft e a Ht e a 5 1 e aS fs t o The above formulas are useful to calculate the Laplace transforms of signals that are defined in a piecewise fashion Chapter 6 Laplace Transharms functions The Dirac delta function or distribution is defined as the limit of the following sequence of narrow top hat functions 1 7 7 Z If ltl e 6 T all at f5 T 0 otherwise Since More generally for a well behaved function g we have gm 6t 7 a dt ga For a gt 0 this allows us to define the Laplace transform of 6t 7 a as f5t dt 1 we also write that 61 dt 1 O 6t7 as e aS Chapter 6 Laplace Transforms Applit Differentiation and integration of Laplace transforms In what follows we write fs as Fs o Differentiation of Laplace transforms tft 5 cit57 5 1 IE5 t 4W 0 Integration of Laplace transforms a 5 0 F1 du r1 F1 11 t 0 Example Find the inverse Laplace transform of ss2 1 Chapter 6 Laplace Transforms rm Applical ODEs and 9m Applications to ODEs and system of ODEs a Solve y y t7r with y7r 0 and y7r117r i 39 a lt lt oLetft 25 Ill tille Whereelt1 Solve 0 otherWIse y 4y 7 5y ft with initial conditions y0 0 and yO 0 Solve y 4y 7 5y 6t71 with initial conditions yw 0 M0 0 d O Solve the initial value problem E AX m an Wig Chapter 6 Laplace Transforms
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