1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

chanter 4

sequ(nces, JUiia

and mandelbrot

sets, and power

• series

Overview

In 1980 Benoit Mandelbrot led a team of mathematicians in producing some
stunning computer graphics from very simple rules for manipulating complex
numbers. This event marked the beginning of a new branch of mathematics,
known as fractal geometry, that has some amazing applications. Many of the
tools needed to appreciate Mandelbrot's work are contained in this chapter. We
look at extensions to the complex domain of sequences and series, ideas that are
part of a standard calculus course.

4.1 Sequences and Series

In formal terms, a complex sequence is a function whose domain is the positive
integers and whose range is a subset of the complex numbers. The following are
examples of sequences:

f (n) = (2- ~) + (s+ ~)i (n= 1, 2, 3, ... );

g(n)= ei"." (n=l,2,3, ... );

h(k) =5+3i+(

1

~i)k (k=l,2,3, ... ); and

(

1 i)n

r(n) =

4

+ Z (n= 1, 2, 3, ... ).

(4-1)

(4-2)

(4-3)

(4-4)

For convenience, at times we use the term sequence rather than complex
sequence. If we want a function s to represent an arbit rary sequence, we can

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