1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

524 CHAPTER 12 • FOURIER SERIES Al'fO THE LAPL ACE TRANSFORM y Figure 12. 12 The Dirichlet problem for the unit disk lzl < 1. ...

12. 2 • THE DIRICHLET PROBLEM FOR THE UNIT D I SK 525 We now turn to the proof Theorem 12.7. The real-valued function 1 - r^2 p ...

526 CHAPTER 12 • FOURlER SERIES AND THE LAPLACE TRANSFORM is known as the Poisson kernel. Expanding the left side of Equation ( ...

12.2 • THE DIRICHLET PROBLEM FOR THE UNIT DISK 527 s s x 0 s = U(I) Figure 12.13 Functions U1 (t) and u1 (1·cos8,rsin8). Using E ...

528 CHAPTER 12 • FOURIER SERIES AND THE LAPLACE TRANSFORM Figure 12.14 U(8)= { 1, 1, -1, tr -8, 8, s for ~ < 8 <tr; fo ...

{ 1, U(O) = 0, 1, 12.3 • VIBRATIONS IN MECHANICAL SYSTEMS 529 for ~ < 8 < 1Ti for - 2 " < 0 < ~; for - 1T<0 & ...

530 CHAPTER 12 • FOURIER SERIES AND THE LAPLACE TRANSFORM --------------------------r------ s = U(t) -~--- ---·-------~· (a) Sys ...

12.3 • VIBRATIONS I N MECHANICAL SYSTEMS 531 We obtain the differential equation for motion by using Newton's second law, which ...

532 CHAPTER 12 • FOURIER SERIES AND THE LAPLACE TRANSFORM Hence the damped mechanical system is governed by the differential equ ...

12.3 • VIBRATIONS IN MECHANICAL SYSTEMS 533 For damped mechanical systems driven by a periodic input F (t), the general solution ...

534 CHAPTER 12 • FOURJER SERI ES AND THE LAPLACE TRANSFORM We obtain the steady st ate solution by assuming that Up (t) has the ...

12.3 • VIBRATIONS IN MECHANICAL SYSTEMS 535 -------~EXERCISES FOR SECTION 12.3 For t hese exercises, use the results of Section ...

536 CHAPTER, 12 • FOURIER SERIES AND THE LAPLACE TRANSFORM Hint: F(t)=:'.!:. 8 +~E. 1 2 cos[(2j-1)t]+~E 1 7r ;=I (2J -1) 7r ; : ...

12.4 • THE FOURIER TRANSFORM 537 where Cn = -^1 1L U (t) e-iwnt/Ldt, 2£ -L for all n. (12-22) We've shown how periodic functions ...

538 CHAPTER 12 • FOURJER SERIES AND THE LAPLACE TRANSFORM ...

12.4 • THE FOURIER TRANSFORM 539 Table 12.1 gives some important properties of the Fourier transform. Linearity Symmetry Time sc ...

540 CHAPTER 12 • FOURI ER SERIES AND THE LAPLACE TRANSFORM Solution Using the result of Example 12.5 and the symmetry property, ...

12.5 • THE LAPLACE TRANSFORM 541 Use the time differentiation property and the result of Exercise 4 to show that -w• ( J° te--Z ...

542 CHAPTER 12 • FOURJER SERJES AND THE LAPLACE TRANSFORM which is commonly referred to simply as the Laplace transform off (t), ...

12.5 • THE LAPLACE TRANSFORM 543 12.5.2 Properties of the Laplace Transform Although a function f ( t) may be defined for all va ...