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# Class Note for MATH 322 with Professor Glickenstein at UA 2

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This 5 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Arizona taught by a professor in Fall. Since its upload, it has received 24 views.

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Date Created: 02/06/15

Chapter 6 Laplace Transforms l Chapter 6 Laplace Transforms Definitions 1 Defiitions o The Laplace transform f of a piecewise continuous function f defined on O7 00 is given by fs Fs 000 eXp s 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 O and k E R such that mm Mexpkt for t 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 71F Chapter 6 Laplace Transforms General properties s shifting Laplace transform of derivatives 34 antiderivatives Definitions Properties of the Laplace transform Applications 1 CDEs and systems of ODES HeaVIside and delta functions rshifting Differentiation and integration of Laplace transforms 2 Properties oaplace transform 0 The Laplace transform is a linear transformation ie if f1 and f2 have Laplace transforms and if a1 and a2 are constants then 5 0411 1 01212 2 11501 04256 0 As for Fourier transforms the statement 1 514 should be understood in a pointwise fashion only at points where f is continuous 0 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 General properties sshifting Laplace transform of derivatives 84 antiderivatives Heaviside and delta funct s tsh ing Differentiation and integr an of Laplace transforms s shifting Laplace transform of derivatives amp antiderivatives 0 Note All of the formulas written in what follows implicitly assume that the various functions used have welldefined Laplace transforms One should therefore check that the corresponding Laplace transforms exist before using these formulas o s shifting formulas c eatfm s Fs a eatfe 1Fs a t 0 Laplace transform of derivatives fs s fs f07 L W s 52 fs s f0 fO Chapter 6 Laplace Transforms General properties sshifting Laplace transform of derivatives 84 antiderivatives HeaVIsicle and delta functions tshifting Differentiation and integration of Laplace transforms Laplace transform of derivatives and antiderivatives Definit39 ns Properties of the Laplace trans m Applications 1 CDEs and systems of ODEs o More generally 5 M s s fs s 1fO s 2fO f 10 o Laplace transform of antiderivatives L Ot r d7 5 fs7 0t f7 d7 71 fsgt S 0 Examples 9 Find the Laplace transforms of sinwt and coswt 0 Find the inverse Laplace transforms of 1552 1 and 15252 Chapter 6 Laplace Transforms General properties sshifting Laplace transform of derivatives amp antiderivatives Heaviside and delta fun tshifting Differentiation and integration of Laplace transforms Haviside and delta functions t shifting 0 The Heaviside function or step function Ht is defined as o iftlt0 Ht 1 iftZO eias 0 We can calculate that for a gt O Ht 2 s s o More generally we have the following timeshifting formulas for a gt 0 cm a Ht 3 s e aS fs ft a Ht a 71 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 Transforms General properties s shifting Laplace transform of derivatives 34 antiderivatives Heaviside and delta functions tshifting Differentiation and integration of Laplace transforms Definitions Properties of the Laplace transform Applications 1 CDEs and systems of ODEs Delta functions 0 The Dirac delta function or distribution is defined as the limit of the following sequence of narrow tophat functions 6tim 22t fEt i iftg sec 0 otherwise 0 Since f6t dt 1 we also write that 6t dt 1 o More generally for a wellbehaved function g we have 00 gt 6t 2 dt ga 700 For a gt 0 this allows us to define the Laplace transform of 6t a as L 6t 2 s e as Chapter 6 Laplace Transforms General properties sshifting Laplace transform of derivatives amp antiderivatives Heaviside and delta functions tshifting Differentiation and integration of Laplace transforms ifferentiation and integration f Laplace transforms In what follows we write fs as Fs o Differentiation of Laplace transforms tft5 FS7 571FSt tft 0 Integration of Laplace transforms c s 0 FV du L 1F1d1gtt 0 Example Find the inverse Laplace transform of ss2 12 Chapter 6 Laplace Transforms Properties of the Laplace 2 form Applications to ODEs and systems of ODEs Applicationto ODEs and systems of ODEs o Solve y y t7r with y7r O and y7r 1 17r 1 Z If 1 6 S t 3 1 6 0 Let for 0 otherwise y 4 5y 2 ft with initial conditions y0 O and yO O o Solve y 4y 5y 2 6t 1 with initial conditions y0 0 M o where 6 lt 1 Solve dX o Solve the initial value problem AX dt Hg ffl Wis Chapter 6 Laplace Transforms

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