1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

244 CHAPTER 6 • COMPLEX INTEGRATION We sometimes state the maximum modulus principle in the following form. EXAMPLE 6.26 Let f ...

6.6 • THE THEOREMS OF MORERA AND LIOUVILLE, AND EXTENSIONS 245 Theorem 6.18 shows that a nonconstant entire function cannot be a ...

246 CHAPTER 6 • COMPLEX INTECRATION EXAMPLE 6.27 Show that the function f (z) = sin z is not a bounded function. Solution We e ...

6.6 • THE THEOREMS OF MORERA AND LJOUVILLE, AND EXTENSIONS 247 t Corollary 6.4 Let P be a polynomial of degree n 2: 1. Then P ca ...

248 CHAPTER 6 • COMPLEX INTEGRATION 5. Let f be analytic in the disk D 5 (0) and suppose that If (z)I ~ 10 for z E C3 (1). (a) F ...

nt Overview Throughout this book we have compared and contrasted properties of complex functions with functions whose domain and ...

250 CHAPTER 7 • TAYLOR AND LAURENT SERIES 00 If Sn (z) is the nth partial sum of the series I:; Ck (z - a/, Statement (7-1) k=O ...

7.1 • UNIFORM CONVERGENCE 251 y 10 8 f<Xo) 6 / ,, ,/ = ======~~-~~~~...- ,, ------- -'=--=--=-"'-"'-"'-"'-=-=-=-=-::".!-!:-:- ...

252 CHAPTER 7 • TAYLOR AND LAURENT SERIES Theorem 7.2 gives an interesting application of the Weierstrass M-test. ...

7. 1 • UNIFORM CONVERCENCE 253 An immediate consequence of Theorem 7.2 is Corollary 7.1. t Corollary 7 .1 For each r, 0 < r & ...

254 CHAPTER 7 • TAYLOR AND LAURENT SERIES 00 i. Corollary 7.2 If the series L: c,.. (z - a)" converges uniformly to f (z) on the ...

7.1 • UNIFORM CONVERGENCE 255 which becomes 00 00 Lo g ( 1 - zo ) = ~ L., --z^1 0 n+l = ~ L., - z^1 0 n. n=O n+l n = l n The p ...

2 5 6 CHAP'l'ER 7 • TAYLOR AND LAURENT SERIES QO Consider the function ((z) = I:; n -', where n - • = exp(-zlnn). n = l (a) Sh ...

2 • T AYLOR SERI ES REPRESENTATIONS 257 Proof ,!,. 0 = (z-a)_:(zo-a) = z~a l - (•o ~)/(z-a). The result now follows from Corol ...

258 CHAPTER 7 • TAYLOR. AND LAUR.ENT SER.JES (7-.6) ...

7.2 • TAYLOR SERIES REPRESENTATIONS 259 A singular point of a function is a point at which the function fails to be analytic. Yo ...

2 6 0 CHAPTER 7 • TAYLOR AND LAURENT SERIES be at least equal to S. We could then make f differentiable at z 0 by redefining f ( ...

7.2 • TAYLOR SERIES REPRESENTATIONS 261 For many students, it makes sense that the first series in Equations (7~14) con- verges ...

262 CHAPTER 7 • TAYLOR AND LAURENT SERIES the identity for sin^3 z , we obtain By the uniqueness of power series, this last expr ...

7 .2 • TAYLOR SERIES REPRESENTATIONS 263 EXAMPLE 7.6 Use the Cauchy product of series to show that 1 00 --- 2 = L (n + 1) z", f ...