Preface ix

Chapter 2 develops the basic properties of complex numbers, with a special em-

phasis on the role of complex conjugation. The author’s own research in complex

analysis and geometry has often used polarization; this technique makes precise the

sense in which we may treat z and z as independent variables. We will view complex

analytic functions as those independent of z. In this chapter we also include pre-

cise deﬁnitions about convergence of series and related elementary analysis. Some

instructors will need to treat this material carefully, while others will wish to re-

view it quickly. Section 5 treats uniform convergence and some readers will wish

to postpone this material. The subsequent sections however return to the basics of

complex geometry. We deﬁne the exponential function by its power series and the

cosine and sine functions by way of the exponential function. We can and therefore

do discuss logarithms and trigonometry in this chapter as well.

Chapter 3 focuses on geometric aspects of complex numbers. We analyze the

zero-sets of quadratic equations from the point of view of complex rather than

real variables. For us hyperbolas, parabolas, and ellipses are zero-sets of quadratic

Hermitian symmetric polynomials. We also study linear fractional transformations

and the Riemann sphere.

Chapter 4 considers power series in general; students and instructors will ﬁnd

that this material illuminates the treatment of series from calculus courses. The

chapter includes a short discussion of generating functions, Binet’s formula for the

Fibonacci numbers, and the formula for sums of p-th powers mentioned above. We

close Chapter 4 by giving a test for when a power series deﬁnes a rational function.

Chapter 5 begins by posing three possible deﬁnitions of complex analytic func-

tion. These deﬁnitions involve locally convergent power series, the Cauchy-Riemann

equations, and the limit quotient version of complex diﬀerentiability. We postpone

the proof that these three deﬁnitions determine the same class of functions until

Chapter 6 after we have introduced integration. Chapter 5 focuses on the relation-

ship between real and complex derivatives. We deﬁne the Cauchy-Riemann equa-

tions using the

∂

∂z

operator. Thus complex analytic functions are those functions

independent of z. This perspective has profoundly influenced research in complex

analysis, especially in higher dimensions, for at least ﬁfty years. We briefly con-

sider harmonic functions and diﬀerential forms in Chapter 5; for some audiences

there might be too little discussion about these topics. It would be nice to develop

potential theory in detail and also to say more about closed and exact diﬀerential

forms, but then perhaps too many readers would drown in deep water.

Chapter 6 treats the Cauchy theory of complex analytic functions in a simpliﬁed

fashion. The main point there is to show that the three possible deﬁnitions of ana-

lytic function introduced in Chapter 5 all lead to the same class of functions. This

material forms the basis for both the theory and application of complex analysis.

In short, Chapter 5 considers derivatives and Chapter 6 considers integrals.

Chapter 7 oﬀers many applications of the Cauchy theory to ordinary integrals.

In order to show students how to apply complex analysis to things they have seen

before, we evaluate many interesting real integrals using residues and contour inte-

gration. We also include sections on the Fourier transform on the Gamma function.