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
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.