1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

124 CHAPTER 4 • SEQUENCES, J ULIA AND MANDELBROT SETS, AND POWER SERIES specify it by writings (1) = z1, s (2) = Z2, and so on. ...

4.1 • SEQUENCES AND SER.JES 125 , • · · ---·'. {All terms z •• for n :;-N,, • :' ~' areinside~D^8 ~). z, , I '' _____-/ . Z2 ,.. ...

126 CHAPTER 4 • SEQUENCES, JULIA AND MANDELBROT SETS, AND POWER SERIES We needed to show the strict inequality lzn -(I < £, a ...

4. 1 • SEQUENCES AND SERIES 127 stipulates that convergent sequences are bounded. The same result holds for complex sequences. A ...

128 CHAPTER 4 • SEQUENCES, JULIA AND MANDELBROT SETS, AND POWER SERIES Definition 4.4: Infinite series 00 The formal expression ...

4. 1 • SEQUENCES ANO SERIES 129 EXAMPLE 4.3 Show that the series f i+ir;f;-^1 r f [;t.z +iC-~t] is n=l n = l convergent. 00 00 ...

130 CHAPTER 4 • SEQUENCES, JULIA AND MANDELBROT SETS, AND POWER SERIES Definition 4.5: Cauchy product 00 00 Let L an and L bn be ...

4.1 • SEQUENCES AND SERIES 1 3 1 EXAMPLE 4.6 Show that f; %"!:~1~ converges. n = l Solution We calculate lznl =I <^35 "!;~ ...

132 CHAPTER 4 • SEQUENCES, JULIA AND MANDELBROT SETS, AND POWER SERIES 00 Show that, if E z., converges, then Jim z,.. = O. Hin ...

4.2 • JULIA AND MANDELBROT SETS 133 quadratic polynomials and indkated his intention to publish a subsequent paper for cubic pol ...

134 CHAPTER 4 • SEQUENCES, JULIA AND MANDELBROT SETS, AND POWER SERIES The complex version of Newton's method also appears to wo ...

4.2 • JULIA AND MANDELBROT SETS 135 front cover of this book illustrate the results of applying our algorithm to various functio ...

136 CHAPTER 4 • SEQUENCES, JULIA AND MANDELBROT SETS, AND POWER SERIES fc- Example 4.9 shows that Ko is the closed unit disk D 1 ...

4.2 • JULIA ANO MANDELBROT SETS 187 The set M has come to be known as the Mandelbrot set. Color plate 6 shows its intricate natu ...

138 CHAPTER 4 • SEQUENCES, JULIA AND MANDELBROT SETS, AND POWER S ERIES Definition 4.7: Attracting point The point ZQ is an attr ...

4.2 • JULIA AND MANDELBROT SETS 139 Definition 4.8: n-cycle An n-cycle for a function l is a set { zo, z 1 , •. • , z,. _ 1} of ...

140 CHAPTER 4 • SEQUENCES, JULIA AND MANDELBROT SETS, AND POWER SERIES (c) Use induction to show that if all the terms of the se ...

4.3 • GEOMETRI C SERIES AND CONVERGENCE THEOREMS 141 4.3 Geometric Series and Convergence Theorems 00 We begin this section by p ...

142 CHAPTER 4 • SEQUENCES, JULIA AND MANDELBROT SETS, AND POWER SERIES 00 t Corollary 4.2 If lzl > 1, the series I: z-n conve ...

4.3 • GEOMETRIC SERIES AND CONVERGENCE T HEOREMS 143 oo n • EXAMPLE 4.14 Evaluate L (~). n =3 Solution We can put this expressio ...