1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

544 CHAPTER 12 • FOURIER SERIES AND THE LAPLACE TRANSFORM formula F (s) might have a domain much larger than this half-plane. La ...

12.5 • T HE LAPLACE TRANSFORM 545 Solution Using the integral definition for .C ( f (t)), we obtain .C (f (t)) = f (t) e- •tdt = ...

546 CHAP'l'ER 12 • FOURIER SERIE.~ AND THE LAPLACE TRANSFORM 1 •EXAMPLE 12.10 Show that .C (t) = 2 · s Solution Integration by p ...

12.5 • THE LAPLACE TRANSFORM 547 Solution Using linearity and lines 6 and 7 of Table 12.2, we obtain c-1 (3s + 6) s^2 + 9 f(t) 1 ...

548 CHAPTER 12 • FOURlER SERIES AND THE LAPLACE TRANSFORM Definition .C (! (t)) = F (s). First derivative .C(f'(t)) = sF(s)-f(O) ...

12.6 • LAPLACE TRANSFORMS OF DERIVATIVES AND INTEGRALS 549 Find .C(2oos4t). 9. Find .C ( e^2 ' -^3 ). Find C (6e- t +3sin 5t). ...

550 CHAPTER 12 • FOURIE R SERIES AND THE LAPLACE TRANSFORM t Corollary 12. 1 If f (t), f' (t) , and f" (t) are of exponential or ...

12.6 • LAPLACE TRANSFORMS OF DERIVATIVES AND INTEGRALS 551 2 6 • EXAMPLE 12.14 Show that .C (t^2 ) = 3 and .C (t^3 ) = 4. s s So ...

552 CHAPTER 12 • FOURIER SERIES AND THE LAPLACE TRANSFORM T he Laplace transform Y (s) of the solution y (t) is easily found to ...

12.7 • SHIITING THEOREMS AND THE STEP FUNCTION 55 3 Find .C(sin^2 t). 4. Show that .C (te') = 1 (s -1)^2. Hint: Let f (t) = te' ...

554 CHAPTER 12 • FOURIER SERIES AND THE LAPLACE TRANSFORM Definition 12.3: Unit step function Let a '2: O. Then, the unit step f ...

12.7 • SHIFTING THEOREMS AND THE STEP FUNCTION 555 y y y = u. (t)ftt-a) 0 a Figure 12.23 Comparison of the functions f (t) and U ...

556 CHAPTER 12 • FOURIER SERIES AND THE LAPLACE TRANSFORM y 1 y =f(t) <>----<> 2 3 Figure 12.24 The function y = f ...

12 .7 • SHIFTING THEOREMS AND THE STEP FUNCTION 5 5 7 1 s We now use Theorem 12.16 and the facts that - and - 2 --are the transf ...

558 CHAPTER 12 • FOURI ER SERIES AND THE LAPLACE TRANSFORM y y = j(t) 0 0 I 2 3 4 5 I Figure 12.25 Let f (t) be as gjven in ...

12.8 • MULTIPLICATION ANO D IVISION BY t 559 12.8 Multiplication and Division by t Sometimes the solutions to nonhomogeneous lin ...

560 CHAPTER 12 • FOURJER SERIES AND THE LAPLACE TRANSFORM 82 - b2 •EXAMPLE12. 21 Showthat.C(tcosbt)= 2. (s2 + b2) S olution If w ...

12.8 • MULTIPLICATION AND D IVISION BY t 561 EXAMPLE 12.23 Use Laplace transforms to solve the initial value problem ty" (t) - ...

562 Cf!APTER 12 • FOURIER SERIES AND THE LAPLACE TRANSFORM Find .C (t^2 cosh t). 7. Show that C (-e' -1) - =In--. s t s - 1 ( ...

12.9 • INVERTING THE LAPLACE TRANSFORM 56 3 ...