Complex Analysis

In this chapter we set the scene and introduce some of the main characters. We begin with the three representations of complex numbers: the Cartesian representation, the polar representation, and the spherical representation. Then we introduce the basic functions encountered in complex analysis: the exponential function, the logarithm function, power functions, and trigonometric functions. We view several concrete functions w = f (z) as mappings from the z-plane to the w-plane, and we consider the problem of describing the inverse functions.

In this chapter we take up the complex differential calculus. After reviewing some basic analysis in Section 1, we introduce complex derivatives and analytic functions in Section 2 and we show that the rules for complex differentiation are the same as the usual rules for differentiation. In Section 3 we characterize analytic functions in terms of the Cauchy-Riemann equations. In Sections 4 and 5 we give several applications of the Cauchy-Riemann equations, to inverses of analytic functions and to harmonic functions. In Section 6 we discuss conformality, which is a direct consequence of complex differentiability. We close in Section 7 with a discussion of fractional linear transformations, which form an important class of analytic functions.

In Sections 1 and 2 we review multivariable integral calculus in order to prepare for complex integration in the next chapter. The salient features are Green’s theorem and independence of path for line integrals. In Section 3 we introduce harmonic functions, and in Sections 4 and 5 we discuss the mean value property and the maximum principle for harmonic functions. Sections 6 and 7 include various applications to physics. The student may proceed directly to complex integration in the next chapter after paging through the review of multivariable calculus in Sections 1 and 2 and reading about harmonic conjugates in Section 3.

In this chapter we take up the complex integral calculus. In Section 1 we introduce complex line integrals, and in Section 2 we develop the complex integral calculus, emphasizing the analogy with the usual one-variable integral calculus. In Section 3 we lay the cornerstone of the complex integral calculus, which is Cauchy’s theorem. The version we prove is an immediate consequence of Green’s theorem. In Section 4 we derive the Cauchy integral formula and use it to show that analytic functions have analytic derivatives. Each of the final four sections features a “named201” theorem. In Section 5 we prove Liouville’s theorem. In Section 6 we give a version of Morera’s theorem that provides a useful criterion for determining whether a continuous function is analytic. Sections 7 and 8, on Goursat’s theorem and the Pompeiu formula, can be omitted at first reading.

In this chapter we show that the analytic functions are exactly the functions that can be expanded in a convergent power series about any point. Since power series can be treated very much as polynomials, this provides a powerful tool for dealing with analytic functions. In Sections 1 and 2 we review infinite series and series of functions. Sections 3 through 6 contain the basic material on power series. In Section 7 we use power series to show that the zeros of an analytic function are isolated. This leads to the uniqueness principle for analytic functions. Section 8 contains a formal definition of analytic continuation, which can be omitted at first reading.

In Section 1 we derive the Laurent decomposition of a function that is analytic on an annulus, and in Section 2 we use the Laurent decomposition on a punctured disk to study isolated singularities of analytic functions. We classify these as removable singularities, essential singularities, or poles, and we characterize each type of singularity. In Section 3 we define isolated singularities at ∞, and in Section 4 we derive the partial fractions decomposition of a rational function. In Sections 5 and 6 we use the Laurent decomposition to study periodic functions and we relate Laurent series to Fourier series. Sections 5 and 6 can be omitted at first reading.

Section 1 is devoted to the residue theorem and to techniques for evaluating residues. In the remaining sections we apply the residue theorem to evaluate various real integrals. This material provides a good training ground for the techniques of complex integration. The student who is anxious to move on can skip the final several sections of the chapter at first reading.

In this chapter we discuss the argument principle and develop several of its consequences. In Section 1 we derive the argument principle from the residue theorem, and we use the argument principle to locate the zeros of analytic functions. Sections 2 through 5 can be viewed as a study of how the zeros of an analytic function depend on various types of parameters. Sections 6 and 7 are devoted to winding numbers of closed paths and the jump theorem for the Cauchy integral. The jump theorem yields an easy proof of the Jordan curve theorem in the smooth case, and a proof of the full Jordan curve theorem is laid out in the exercises. In Section 8 we introduce simply connected domains and we characterize these in several ways. While the material in this chapter is of fundamental importance for the Riemann mapping theorem in Chapter XI and for various further developments, the student can skip to Chapter IX immediately after Sections 1 and 2.

This short chapter is devoted to the Schwarz lemma, which is a simple consequence of the power series expansion and the maximum principle. The Schwarz lemma is proved in Section 1, and it is used in Section 2 to determine the conformal self-maps of the unit disk. In Section 2 we formulate the Schwarz lemma to be invariant under the conformal self-maps of the unit disk, thereby obtaining Pick’s lemma. This leads in Section 3 to the hyperbolic metric and hyperbolic geometry of the unit disk.

In Section 1 we introduce the Poisson kernel function and we develop the Poisson integral representation for harmonic functions on the open unit disk. The Poisson kernel is the analogue for harmonic functions of the Cauchy kernel for analytic functions, and the Poisson integral formula solves the Dirichlet problem for the unit disk. In Section 2 we use this solution to characterize harmonic functions by the mean value property. This characterization is the analogue of Morera’s theorem characterizing analytic functions. In Section 3 we apply the characterization of harmonic functions to establish the Schwarz reflection principle for harmonic functions. The reflection principle plays a key role in the study of boundary behavior of conformal maps.

In this chapter we will be concerned with conformal maps from domains onto the open unit disk. One of our goals is the celebrated Riemann mapping theorem: Any simply connected domain in the complex plane, except the entire complex plane itself, can be mapped conformally onto the open unit disk. We begin in Section 1 by reviewing and enlarging our repertoire of conformal maps onto the open unit disk, or equivalently, onto the upper half-plane. In Section 2 we state and discuss the Riemann mapping theorem. Before embarking on the proof, we give some applications to the conformal mapping of polygons in Section 3 and to fluid dynamics in Section 4. In Section 5 we develop some prerequisite material concerning compactness of families of analytic functions, which is at a deeper level than the analysis used up to this point. The proof of the Riemann mapping theorem follows in Section 6.

In Sections 1 and 2 we treat normal families of meromorphic functions. These are families that are sequentially compact when regarded as functions with values in the extended complex plane. We give two characterizations of normal families, Marty’s theorem in Section 1 and the Zalcman lemma in Section 2. From the latter characterization we deduce Montel’s theorem on compactness of families of meromorphic functions that omit three points, and we also prove the Picard theorems. Sections 3 and 4 constitute an introduction to iteration theory and Julia sets. In Section 3 we proceed far enough into the theory to see how Montel’s theorem enters the picture and to indicate the fractal nature (self-similarity) of Julia sets. In Section 4 we relate the connectedness of Julia sets to the orbits of critical points. In Section 5 we introduce the Mandelbrot set, which has been called the “most fascinating and complicated subset of the complex plane.”

In this chapter we prove two fundamental theorems, one “additive” and the other “multiplicative,” on prescribing zeros, and poles of meromorphic functions. The first is the Mittag-Leffler theorem, which asserts that we can prescribe the poles and principal parts of a meromorphic function. The second is the Weierstrass product theorem, which asserts that we can prescribe the zeros and poles, including orders, of a meromorphic function. The theorems are closely related. Both theorems are proved by the same type of approximation procedure, which depends on Runge’s theorem on approximation by rational functions. We prove Runge’s theorem in Section 1, followed by the Mittag-Leffler theorem in Section 2. In Section 3 we introduce infinite products, which can always be converted to infinite series by taking logarithms. We prove the Weierstrass product theorem in Section 4.

Our aim in this chapter is to illustrate the power of complex analysis by proving a deep theorem in number theory, the prime number theorem, which does not appear at first glance to be related to complex analysis. Along the way we introduce various functions that play an important role in complex analysis. In Section 1 we introduce the gamma function Γ(z), which provides a meromorphic extension of the factorial function. We derive the asymptotic properties of the gamma function in Section 2 by viewing it as a Laplace transform. This yields Stirling’s asymptotic formula for n!. In Section 3 we study the zeta function, which is a meromorphic function whose zeros are related to the asymptotic distribution of prime numbers. In Section 4 we study Dirichlet series associated with various number-theoretic functions, thereby giving a strong hint of the fecund relationship between complex analysis and number theory. The proof of the prime number theorem is given in Section 5.

In Chapter X we used the Poisson kernel to solve the Dirichlet problem for the unit disk. In this chapter we study the Dirichlet problem for more general domains in the plane. The basic method, due to O. Perron, is to look for the solution of the Dirichlet problem as the upper envelope of a family of subsolutions. In Section 2 we introduce subharmonic functions, which play the role of the subsolutions. In Section 3 we derive Harnack’s inequality, which provides a compactness criterion for families of harmonic functions. Perron’s procedure for solving the Dirichlet problem is developed in Section 4. We apply the method to give another proof of the Riemann mapping theorem in Section 5. In Sections 6 and 7 we introduce Green’s function.

Our goal in this chapter is to prove the uniformization theorem for Riemann surfaces and to indicate its usefulness as a tool in complex analysis. We begin in Sections 1 and 2 by defining Riemann surfaces, providing examples, and showing how various local notions as analytic function, meromorphic function, and harmonic function carry over to Riemann surfaces. In Sections 3 and 4 we define Green’s function for Riemann surfaces and show that Green’s function is symmetric. In Section 5 we show that every Riemann surface has bipolar Green’s functions. We prove the uniformization theorem in Section 6. The proof depends on Green’s function when it exists and on bipolar Green’s function otherwise. In Section 7 we define covering spaces and covering maps, and we state several results that indicate the power of the uniformization theorem.