1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

144 CHAPTER 4 • SEQUENCES, JULIA AND MANDELBROT SETS, AND POWER SERIES • EXAMPLE 4.16 Show that the series f: (z;:,!)" converges ...

4.3 • GEOMETRIC $ERIES AND CONVERGENCE THEOREMS 1 4 5 in the sequence larger than L +t:, b ecause L+t: < 3, as the following ...

146 CHAPTER 4 • SEQUENCES, JULIA AND MANDELBROT SETS, AND POWER SERIES Note that, in applying either Theorem 4.13 or 4.14, if L ...

4 .4 • POWER SERIES FUNCTI ONS 147 Est ablish t he claim in the proof of Theorem 4.12 that if lzl < 1, then n -Jim oo z" = ...

148 CHAPTER 4 • SEQUENCES, JULIA AND MANDELBROT SETS, AND POWER SERIES Another way to phrase case (ii) of Theorem 4.15 is to say ...

4.4 • POWER $ERIES FUNCTIONS 149 y Divergence What happens on the boundary may be unknown. Figure 4.3 The radius of convergence ...

150 CHAPTER 4 • SEQUENCES, JULI A AND MANDELBROT SETS, AND POWER SERIES We come now to the main result of this section. ...

4 .4 • POWER SERIES FUNCTI ONS 151 ...

1 5 2 CHAPTER 4 • SEQUENCES, JULIA AND MANDELBROT SETS, AND POWER SERIES y -+-------------- x Figure 4.4 Choosing 8 to prove tha ...

4.4 • POWER SERIES FUNCTIONS 153 and termwise differentiation shows that its derivative is We leave as an exercise to show that ...

154 CHAPTER 4 • SEQUENCES, JULIA AND MANDELBROT SETS, AND POWER SERIES 00 (i) g (z) = L: z^2 ". n=O 00 n (j) g (z) = L: !!..,z". ...

c. a ter 5 e em -ntary functions Overview How should cornplex-valued functions such as ez, log z, sin z, and the like, be define ...

156 CHAPTElt 5 • ELEMENTARY FUNCTIONS Clearly, this definition agrees with that of the real exponential function when z is a rea ...

5.1 • THE COMPLEX EXPONENTIAL FUNCTION 157 Note that parts (ii) and (iii) of the Theorem 5.1 combine to verify DeMoivre's fonnul ...

158 CHAPTER 5 • ELEMENTARY FUNCTIONS y v 41t 3 21t 1t ". .. . 2 - 1 l • 2 x Figure 5.1 T he points {zn} in the z plane (i.e., ...

5.1 • THE COMPLEX EXPONENTIAL FUNCTION 159 Thezplane. w=el- -+- x v rj .. Thew plane. Figure 5.2 The fundamental period strip fo ...

160 CHAPTER 5 • ELEMENTARY FUNCTIONS v w = exp(z) y x " The zplane. Thew plane. Figure 5.3 The image of R under the transformati ...

5.1 • THE COMPLEX EXPONENTIAL FUNCTION 161 (c) Precisely where should the images of the points ±i7r be located? Verify Equation ...

162 CHAPTER 5 • ELEMENTARY FUNCTIONS Let n be a positive integer. Show that (a) (expz)" =exp (nz). (b) (•x;z)" = exp (-nz). 00 ...

5 .2 • THE COMPLEX LOGARITHM 163 (e) Use part (d} to show that p" (y) + p (y) = 0 and q" (y) + q (y) = O. (f) Identify the gener ...