Chapter 7 Taylor and Laurent Series
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 formulas. However, there are 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 defined on the real numbers to be differentiable everywhere and yet not be expressible as a power series (see Exercise 7.2.20 of 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 \(D_r(\alpha)\text{,}\) its Taylor series about \(\alpha\) converges to the function at every point in this disk. Thus, analytic functions are locally nothing more than glorified polynomials.
