1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

264 CHAPTER 7 • TAYLOR AND LAURENT SERIES -------... EXERCISES FOR SECTION 7.2 l. By computing derivatives, find the Maclaurin s ...

7.2 • TAYLOR SERIES REPRESENTATIONS 265 Find f <^3 > (0) for 00 (a) f (z) = L: (3 + ( - 1)")" z". n=O (b) g (z) = E < ...

266 CHAPTER 7 • TAYLOR AND LAURENT SERIES ( c) Use the partial sum involving terms up to z^9 to find approximations to C (1.0) a ...

7.3 • LAURENT SERIES REPRESENTATIONS 267 7.3 Laurent Series Representations Suppose that f (z) is not analytic in DR (a) but is ...

268 CHAPTER 7 • TAYLOR AND LAURENT SERIES Definition 7 .4: Annulus Given 0 Sr < R, we define the annulus centered at O' with ...

7.3 • LAURENT SERIES REPRESENTATIONS 269 The main result of this section specifies how functions analytic in an annulus can be e ...

270 CHAPTER 7 • TAYLOR AND LAURENT SERIES . (7-25). (7-26) 1, and we can. tise the (7-27) .. (7-28) ...

7.3 • LAURENT $ERIES REPRESENTATIONS 271 What happens to the Laurent series if f is analytic in the disk Dn (a)? Look- ing at Eq ...

272 CHAPTER 7 • TAYLOR AND LAURENT SERIES series in Equation (7-22) involving the positive powers of (zo -a) is actually the Tay ...

7.3 • LAURENT SERIES REPRESENTATIONS 273 Equation (7-23). The following examples illustrate some methods for finding Laurent ser ...

274 CHAPTER 7 • TAYLOR AND LAURENT SERIES • EXAMPLE 7.8 Find the Laurent series representation for f (z) =^00 ',.!-^1 that invol ...

7.3 • LAURENT SERIES REPRESENTATIONS 275 9. Find two Laurent series for z-^1 (4 - z) -^2 involving powers of z and state where t ...

276 CHAPTER 7 • TAYLOR AND LAURENT SERIES The Z-transform. Let {an} be a sequence of complex numbers satisfying the growth cond ...

7.4 • SINGULARITIES, ZEROS, AND POLES 277 A (a, 0, R). We now look at this special case of Laurent's theorem in order to classif ...

278 CHAPTER 7 • TAYLOR AND LAURENT SERIES Another example is g (z) = co•.1-^1 , which bas an isolated singularity at the point 0 ...

7.4 • $INGULARITIES, ZEROS, AND POLES 279 EXAMPLE 7.10 From Theorem 7.10 we see that the function z1 z11 z15 f (z) = zsin z^2 ...

280 CHAPTER 7 • TAY1-0R AND LAURENT SERIES An immediate oonsequence of Theorem 7. 11 is Corollary 7.4. The proof is left as a.n ...

7.4 • SINGULARITIES, ZEROS, AND POLES 281 Corollaries 7.5- 7.8 are useful in determining the order of a zero or a pole. The proo ...

282 CHAPTER 7 • TAYLOR AND LAURENT SERIES Corollary 7.6 If f has a pole of order k at the point a, then g (z) = 7h> has a r ...

7.4 • SJNGULARJTIES, ZEROS, AND POLES 283 •EXAMPLE 7.14 Locate the poles of g(z) = "co;~ul and specify their order. Solution The ...