1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

284 CHAPTER 7 • TAYLOR AND LAURENT SERIES (k) (1 - expz)-^1 • (1) z-^6 sinh z. L-0cate t he singularities of the following func ...

7.5 • APPLICATIONS OF TAYLOR AND LAURENT SERIES 285 7.5 Applications of Taylor and Laurent Series. In this section we show how y ...

286 CHAPTER. 7 • TAYLOR. AND LAURENT SERIES Theorem 7.13 also allows us to give a simple argument for one version of L'Hopital's ...

7 .5 • APPLICAT IONS OF TAYLOR AND L AURENT SERIES 287 Solution Using long division, we see that 1 1 12 54 secz = --= 2 • • = 1 ...

288 CHAPTER 7 • TAYLOR AND LAURENT SERIES II Corollary 7.12 If f is analytic in D; (a), then f can be defined to be analytic at ...

7.5 • APPLICATI ONS OF TAYLOR AND LAURENT SERIES 289 Laurent series for g (z) in the annulus D; (0) is 00 (-1)" 1 g(z) = 1 + 2:: ...

290 CHAPTER 7 • TAYLOR AND LAURENT SERIES 6. Use L'Hopital's rule to find the following limits. (a) z-1+i lim ~-z +4 (b) I. x (^ ...

Overview You now have the necessary machinery to see some amazing applications of the tools we developed in the last few chapter ...

292 CHAPTER 8 • RESIDUE THEORY • EXAMPLE 8.1 If f (z) = exp(~), then the Laurent series off about the point 0 has the form ( 2) ...

8.1 • THE RESIDUE THEOREM 293 • EXAMPLE 8.3 Evaluate f exp(;) dz. ct(o) Solution In Example 8.1 we showed that the residue of f ...

294 CHAPTER 8 • RESIDUE THEORY Figure 8.1 The domain D and contour C and the singular points z 1 , zz, ... , z,. in the statemen ...

8.1 • THE RESIDUE THEOREM 295 EXAMPLE 8 .4 Find the residue off (z) = " <^0 ;<"•) at zo = O. Solution We write f (z) = ; ...

296 CHAPTER 8 • RESIDUE THEORY This last limit involves an indeterminate form, which we evaluate by using L'Hopital's rule: Res ...

8. 1 • THE RESIDUE THEOREM 297 If z 0 is any one of the singularities of f, then we can use L'Hopital 's rule to compute Reslf,z ...

298 CHAPTER 8 • RESIDUE THEORY which is valid for lz - al < le - al. Thus, the Laurent series off about the point a is A 00 [ ...

8.1 • THE RESIDUE THEOREM 299 where . z^2 + 3z + 2 A = Res[zf(z),O] = hm = - 2, 0 z - 1 B = Res[f,O] =Lim d zZ +^3 z +^2 • ...

300 CHAPTER 8 • RESIDUE THEORY Let f and g have an isolated singularity at z 0. Show that Res[/+ g, zo) = Res[f , z o] + Res(g, ...

8.2 • TRIGONOMETRIC INTEGRALS 301 Find J (3z^4 + 10z^2 + 3)- 1 dz when c (a) c =ct (iv'3). (b) C= Ct(~)· 8. Find J (z^4 - z^3 ...

302 CHAPTER 8 • RESIDUE THEORY Suppose that we want to evaluate an integral of the form {2" lo F (cosO,sinO) dO, (8-3) where F ( ...

z=cos 9 +isin9 -4-<------>--<~ 9 - 0 2x (a) The interval [0, 2x] of integration for F(oos 9, sin (/). 8.2 • TRIGONOMETR ...