nt
Overview
Throughout this book we have compared and contrasted properties of complex
functions with functions whose domain and range lie entirely within the real
numbers. There are many similarities, such as the standard differentiation for-
mulas. However, there a.re also some surprises, and in this chapter you will
encounter one of the hallmarks that distinguishes complex functions from their
real counterparts: It is possible for a function define<! on the real numbers to be
differentiable everywhere and yet not be expressible as a power series (see Exer-
cise 20 , Section 7.2). For a complex function, however, things are much simpler!
You will soon learn that if a complex function is analytic in the disk Dr (a), its
Taylor series about a converges to the function at every point in this disk. Thus,
analytic functions are locally nothing more than glorified polynomials.
7. 1 Uniform Convergence
Complex functions are the key to unlocking many of the mysteries encounteredwhen power series are first introduced -in· a ·calculus course. We begin by dis-
cussing an important property associate<! ·with power series-uniform conver-
gence.
Recall that, for a function f defined on a set T , the sequence of funct ions{Sn} converges to fat the point zo ET, provided lim Sn (zo) = f (zo). Thus,
n~oo
for the particular point zo, we know that for each c > 0, there exists a positive
integer Ne,zo (depending on both c and zo) such thatif n ;::: N,,. 0 , then !Sn (zo) -f (zo)I < c. (7-1)
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