1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

564 CHAPTER 12 • FOURIER SERIES AND THE LAPLACE TRANSFORM s^3 -4s+l. • EXAMPLE 12.24 Let Y (s) = 3. Find c-^1 (Y (s)). s(s- 1) S ...

12.9 • INVER.TING THE LAPLACE TRANSFORM 565 • EXAMPLE 12. 25 Let Y (s) = (s 2 + 4 ~~ 82 + 9 ). Find e,-t (Y (s)). Solution Here ...

566 CHAPTER 12 • FOURIER SERIES AND THE LAPLACE TRANSFORM of Q (s) occur at 0 ± 2i and 0 ± 3i. Computing the residues yields Res ...

12.9 • I NVERTING THE LAPLACE TRANSFORM 567 and we obtain A = 4 and B = - 4. Therefore, 1 -2 - 2 2l(s- 0)- 2(- l)(1) Y(s)=-+--+ ...

568 CHAPTER 12 • FOURIER SERIES AND THE LAPLACE TRANSFORM According to Equation (12-29}, the inverse Laplace transform is given ...

12.9 • INVER.TlNG THE LAPLACE TR.ANSFORM 5 6 9 Figure 12. 27 The contour f R· EXAMPLE 12. 28 Find the inverse Laplace transform ...

570 CHAPTER 12 • FOURIER. $ERIES AND THE LAPLACE TRANSFORM P(-2) -8+ 3 use Q' (s) = 3s^2 + 4s + 1, calculation reveals that Q' ( ...

U. y (s) = s 3 3s 2 s + 1. s^5 - s 3 2 3 y (s) = s + s + s +. g5 - s .. s^3 + 2s^2 + 4s + 2 Fmd the mverse of Y (s) = ( 2 ) ...

572 CHAPTER 12 • FOURIER SERIES AND THE LAPLACE TRANSFORM ...

12.10 • CONVOLUTION 573 Figure 12.28 The region of integration in the convolution theorem. Table 12 .4 lists the properties of c ...

574 CHAPTER 12 • FOURI ER SERIES AND T HE LAPLACE TRANSFORM Solution Letting F(s) = .C(f (t)) and using .C(t) = 1 s^2 in the con ...

12.10 • CONVOLUTION 575 y 100 80 60 40 20 y= o, (t) ill 100 80 y y 100 80 40. 20. 1 1= 01 <•! 100 Figure 12 .29 Graphs of y ...

576 CHAPTER 12 • FOURIER SERIES AND THE LAPLACE TRANSFORM y 0.4 0.2 --0.5 -0.I Figure 12.31 The solution y = y (t). Solution Tak ...

12. 10 • CONVOLUTION 577 EXAMPLE 12.33 Use the convolution method to solve the !VP y^11 (t) + y (t) = tan t with y (0) = 1 and ...

57 8 C HAPTER 12 • FOURIER SERIES AND THE LAPLACE TRANSFORM Therefore, the solution is cost y(t) =u(t)+v(t) = cost+3sint+cos(t)l ...

12 .10 • CONVOL UTION 579 Find .C (J; e-T cos (t - r) dr). 16. Find .C (l; (t - r)^2 e^7 dr). Let F(s) = .C(f(t)). Use convol ...

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Answer s Section 1 .1. The Origi n of Complex Numbers: page 6 1. Mimic the argument the text gives in showing 2 + A = V2 + ./-12 ...

S82 ANSWERS Sa. Since z 1 is a root of the polynomial P, P(zi) = O. Use properties (1-12) through (1-14) of Theorem 1.1 to show ...

ANSWERS 583 By the triangle inequality, lz1 - z2I = lz1 + (-z2)I :S lz1l+[-.z:i] = lz1l+lz2I · Let z =(a, b). Then z =(a, -b), ...