1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

324 C HAPTER 8 • RESIDUE THEOltY EXAMPLE 8.23 Evaluate P.V. Jo"'° x(::i) dx, where 0 <a< 1. Solution The function f (z) ...

8.6 • INTEGRANDS WITH BRANCH POINTS 325 Using Theorem 8.7, we have dx= Res -1 = - 1 (^00) x" 2 7ri 27ri ( e;,,,, ) 0 x (x + 1) 1 ...

326 CHAPTER 8 • REsIDUE THEORY Keeping in mind the branch of logarithm that we're using, we then have fct(z)dz= 1: f(x)dx+ 1-c. ...

7 • THE ARGUMENT PRINCIPLE AND ROUCHE'S THEOREM 327 oo x' dx 3 .P.V.fo 2· (1 + x) l oo X'l dx P.V. fo -1 2 · x p roo In (x^2 ...

328 CHAPTER 8 • RESIDUE THEORY Definition 8 .3: Meromorphic function A function f is said to be m eromorphic in a domain D provi ...

8.7 • THE ARGUMENT PRINClPLE AND RoUCHE'S'THEOREM 329 t Corollary 8.1 Suppose that f is analytic in the simply connected domain ...

330 CHAPTER 8 • RESIDUE THEORY y Figure 8.10 The points Zk on the contour C that winds around z*. change by 27r (Z1 - P1) radian ...

8.7 • THE ARGUMENT PRI NCIPLE AND ROUCHE'S THEOREM 331 v Figure 8.11 The points Wk on the contour f ( C) that winds a.round 0. t ...

332 CHAPTER 8 • RESIDUE T HEORY v Figure 8. 12 The image curve f (C2 (O)) under f (z) = z^2 + z. radian measure off ( C) around ...

8.7 • THE ARGUMENT PRINGIPLE AND R.oUCHE'S THEOREM 333 Remark 8.4 Letting f (z) = z in Theorem 8.9 gives W(C,a)= -^1 1 1 { 1 2 . ...

334 CHAPTER 8 • REsIDUE THEORY t Corollary 8. 2 Suppose that f and g a.re analytic functions defined in the simply connected dom ...

7 • THE ARGUMENT PRINCIPLE AND RoucHii:'s THEOREM 335 Solution Let f (z) = 7z + l, then f (z) + g (z) = z^4. At points on the ...

336 CHAPTER 8 • RESIDUE THEORY Let g (z) = z^6 - 5z^4 + 10. (a) Show that there are no zeros in lzl < 1. (b) Show that ther ...

~er 9 and sf-orms ations Overview The z-transform is useful for the manipulation of discrete data sequences and has acquired a n ...

338 CHAPTER, 9 • Z-TRANSFORMS AND APPLICATIONS Remark 9.1 The z-transform is defined at points z EC where the Laurent series (9- ...

9.1 • THE Z-TR.ANSFOR.M 339 9.1.1 Admissible Form of a z-transform Formulas for X(z) do not arise in a vacuum. In an introductor ...

340 CHAPTER 9 • Z· TRANSFORMS AND APPLICATIONS z-transform, provided that it is a. rational function; that is, X(z) = P(z) = bo ...

1 • THE Z-TRANSFORM 341 Solution This follows trivially from Equation (9-1) X(z) = 3[xn) = L:=O XnZ-n = 1 + L::'=l Qz- n = 1. ...

342 CHAPTER 9 • Z·TRANSFORMS AND APPLICATIONS Solution Remark 9.4 When using the residue theorem to compute inverse z·transforms ...

9.1 • THE Z-TRANSFORM 343 •EXAMPLE 9.6 Find the inverse z-transform Xn = x [n] = 3 -^112 ;:_ 1 ]. Use (a) series, (b) the table ...