A Course in Complex Analysis

The fundamental concept of holomorphic function is introduced via complex differentiability in section I.1. The relation between real and complex differentiability is then discussed, leading to the characterization of holomorphic functions by the Cauchy-Riemann differential equations (I.2). Power series are important examples of holomorphic functions (I.3); we here apply real analysis to show their holomorphy, although Chapter II will open a simpler way. In particular, the real exponential and trigonometric functions can be extended via power series to holomorphic functions on the whole complex plane; we discuss these functions without recurring to the corresponding real theory (I.4). Section I.5 presents an essential tool of complex analysis, viz. integration along paths in the plane. In I.6 we carry over the basic theory to functions of several complex variables.

Holomorphic functions differ fundamentally from real differentiable functions: they are infinitely often (real and complex) differentiable (II.3, II.7), they even admit power series expansions (II.4), their entire behaviour is determined by their values on arbitrarily small open sets (II.4, II.7), and they satisfy powerful convergence theorems and estimates (II.5). All of these properties are consequences of the Cauchy integral theorem and the integral representations that arise from it (II.1–3). Meromorphic functions extend the class of holomorphic functions (II.6); their study leads to the notion of isolated singularities and to generalizations of power series obtained by allowing negative powers (Laurent series). In addition to the phenomena that occur in the theory of functions of one complex variable, a fundamentally new phenomenon enters the picture in higher dimensions: the simultaneous holomorphic continuation of all holomorphic functions from a given domain to a larger one (II.7). Here the Cauchy integral formula (in one variable!) is again the decisive tool.

By adding a “point at infinity”, denoted ∞, to the complex plane, we obtain the Riemann sphere (III.1); it allows an elegant description of meromorphic and, in particular, rational functions and an interpretation of Möbius transformations as automorphisms of the sphere (III.2,4). Important theorems about functions that are holomorphic on all of c (“entire functions”) follow from the fact that the point ∞ is an isolated singularity of these functions (III.3). Polynomials and rational functions are investigated in detail in III.2; in particular, this section contains proofs of the fundamental theorem of algebra as well as historical notes. With the logarithm function and the functions that arise from it, we conclude our “elementary” study of the elementary functions; among other things, we describe the local mapping properties of holomorphic or meromorphic functions via root functions. Partial fraction decompositions (III.6) are an essential tool in the study of meromorphic functions; in addition to a general existence theorem, this section contains the decompositions of the functions cot πz and 1/ sin πz and their consequences. The Weierstrass product formula (III.7) for entire functions substantially generalizes the factorisation of polynomials into linear factors. We shall use it in V.1 and V.4 to define non-elementary functions.

The concept of the winding number allows a general formulation of the Cauchy integral theorems (IV.1), which is indispensable for everything that follows. IV.2 presents a generalization of the Cauchy integral formula to real differentiable functions; it will play a basic role in Chapter VI. With the Laurent series expansion (IV.3) and the residue theorem (IV.4), further essential tools of complex analysis are at our disposal. They will be used to evaluate complicated integrals (IV.5) and then to study the equation f(z) = w, where f is a holomorphic function (IV.6). If one makes the integral formulas from sections IV.3 and IV.6 dependent on parameters, then one obtains the Weierstrass preparation theorem (IV.7), which gives fundamental information about the zeros of holomorphic functions of several variables.

This chapter presents the analogues of the Mittag-Leffler and Weierstrass theorems for functions of several complex variables. To this end it develops fundamental methods of multivariable complex analysis that reach far beyond the applications we are going to give here. – Meromorphic functions of several variables are defined as local quotients of holomorphic functions (VI.2); the definition requires some information on zero sets of holomorphic functions (VI.1). After introducing principal parts and divisors we formulate the main problems that arise: To find a meromorphic function with i) a given principal part (first Cousin problem) ii) a given divisor (second Cousin problem); iii) to express a meromorphic function as a quotient of globally defined holomorphic functions (Poincaré problem). These problems are solved on polydisks – bounded or unbounded, in particular on the whole space – in VI.6–8. The essential method is a constructive solution of the inhomogeneous Cauchy-Riemann equations (VI.3 and 5) based on the one-dimensional inhomogeneous Cauchy formula – see Chapter IV.2. Along the way, various extension theorems for holomorphic functions are proved (VI.1 and 4). Whereas the first Cousin problem can be completely settled by these methods, the second requires additional topological information which is discussed in VI.7, and for the Poincaré problem one needs some facts on the ring of convergent power series which we only quote in VI.8.

We study holomorphic, in particular biholomorphic, maps between domains in c and, in one case, in c ⁿ , n > 1. These maps are for n = 1 conformal (angle and orientation preserving); so we shall use the terms biholomorphic and conformal interchangeably in this case. For n > 1 we consistently use biholomorphic. Automorphisms of domains, i.e. biholomorphic self-maps, are determined for disks resp. half-planes, the entire plane, and the sphere: they form groups consisting of Möbius transformations (VII.1). The proof of this fact relies on an important growth property of bounded holomorphic functions: the Schwarz lemma 1.3. Because the automorphism group of the unit disk (or upper half plane) acts transitively, it gives rise – according to F. Klein's Erlangen programme – to a geometry, which turns out to be the hyperbolic (non-euclidean) geometry (VII.2 and 3). The unit disk is conformally equivalent to almost all simply connected plane domains: Riemann's mapping theorem, proved in VII.4. For n > 1 even the immediate generalisations of the disk – the polydisk and the unit ball – are not biholomorphically equivalent (VII.4). Riemann's mapping theorem can be generalised to the general uniformization theorem (VII.4): a special but exceedingly useful case of this is the modular map λ which we introduce in VII.7. Its construction uses tools that are also expedient for other purposes: harmonic functions (with a solution of the Dirichlet problem for disks) and Schwarz's reflection principle (VII.5 and 6). The existence of λ finally yields two important classical results: Montel's and Picard's “big theorems”.