Theory of Complex Functions

... „Zuvorderst wiirde ich jemand, der eine neue Function in die Analyse einführen will, urn eine Erklärung bitten, ob er sie schlechterdings bloss auf reelle Grössen (reelle Werthe des Arguments der Function) angewandt wissen will, und die imaginären Werthe des Arguments gleichsam nur als ein Überbein ansieht - oder ob er meinem Grundsatz beitrete, dass man in dem Reiche der Grössen die imaginären a + b\( % GHsislcaaIXaaaleqaaaaa!37B7! \sqrt { - 1} \) = a + bi als gleiche Rechte mit den reellen geniessend ansehen müsse. Es ist hier nicht von praktischem Nutzen die Rede, sondern die Analyse ist mir eine selbständige Wissenschaft, die durch Zurücksetzung jener fingirten Grössen ausserordentlich an Schönheit und Rundung verlieren und alle Augenblick Wahrheiten, die sonst allgemein gelten, höchst lästige Beschränkungen beizufügen genöthigt sein würde ... (At the very beginning I would ask anyone who wants to introduce a new function into analysis to clarify whether he intends to confine it to real magnitudes (real values of its argument) and regard the imaginary values as just vestigial - or whether he subscribes to my fundamental proposition that in the realm of magnitudes the imaginary ones a + b\( % GHsislcaaIXaaaleqaaaaa!37B7! \sqrt { - 1} \) = a + bi have to be regarded as enjoying equal rights with the real ones. We are not talking about practical utility here; rather analysis is, to my mind, a self-sufficient science. It would lose immeasurably in beauty and symmetry from the rejection of any fictive magnitudes. At each stage truths, which otherwise are quite generally valid, would have to be encumbered with all sorts of qualifications ... ).“

The adage leading off this chapter is the kernel of all differential calculi. Notwithstanding that this cornerstone was pulverized by Riemann’s and Weierstrass’ discovery of (real-valued) everywhere continuous nowhere differentiable functions on ℝ, it is still a valuable principle for creative mathematicians and physicists.

The investigation of length-preserving, respectively, angle-preserving mappings between surfaces in ℝ³ is one of the interesting problems addressed in classical differential geometry. This problem is important for cartography: every page of an atlas is a mapping of apart of the (spherical) surface of the earth into a plane. We know that there cannot be any length-preserving atlases; but by contrast there are indeed angle-preserving atlases (e.g., those based on stereographic projection). The first goalofthis chapter is to show that for domains in the plane ℝ² = ℂ angle-preserving mappings and holomorphic functions are essentially the same thing (Section 1). The interpretation of holomorphic functions as angle-preserving (= conformal) mappings was advocated especially by Riemann (cf. 1. 5).It provides the best way to “intuitively comprehend” such functions. One examines in detail how paths behave under such mappings. The invariance, under the mapping, of the angles in which curves intersect each other frequently makes possible a good description of the function. “The conformal mapping associated with an analytic function affords an excellent visualization of the properties of the latter; it can well be compared to the visualization of areal function by its graph” (Ahlfors [1], p. 89).

Outside of the polynomials and rational functions, which arise from applying the four basic species of calculation finitely often, there really aren’t any other interesting holomorphic functions available at first. Further functions have to be generated by (possibly multiple) limit processes; thus, for example, the exponential function exp z is the limit of its Taylor polynomials \( % ggHiLdGccaGGVaGaamODaiaacgcaaaa!3E1C! \sum\nolimits_0^n {{z^v}} /v!\) or, as well, the limit of the Euler sequence (1 + z / n)ⁿ . The technique of getting new functions via limit processes was described by GAUSS as follows (Werke 3, p.198): “Die transscendenten Functionen haben ihre wahre Quelle allemal, offen liegend oder versteckt, im Unendlichen. Die Operationen des Integrirens, der Summationen unendlicher Reihen... oder überhaupt die Annäherung an eine Grenze durch Operationen, die nach bestimmten Gesetzen ohne Ende fortgesetzt werden - dies ist der eigentliche Boden, auf welchem die transscendenten Functionen erzeugt werden...” (The transcendental functions all have their true source, overtly or covertly, in the infinite. The operation of integration, the summation of infinite se ries... or generally the approach to a limit via operations which proceed according to definite laws but without termination - this is the real ground on which the transcendental functions are generated... )

The series of functions which are the most important and fruitful in function theory are the power series, series which as early as 1797 had been considered by LAGRANGE in his Théorie des fonctions analytiques. In this chapter the elementary theory of convergent power series will be discussed. This theory used to be known also as algebraic analysis (from the subtitle Analyse algébrique of Cauchy’s Cours d’analyse [C]). Also of interest in this connection is the article of the same title by G. Faber and A. Pringsheim in the Encyklopädie der Mathematischen Wissenschaften II, 3.1, pp. 1–46 (1908).

In this chapter the classical transcendental functions, already treated by Euler in his Introductio [E], will be discussed. At the center stands the exponential function, which is determined (§1) both by its differential equation and its addition theorem. In Section 2 we will prove directly, using differences and the logarithmic series and without borrowing any facts from real analysis, that the exponential function defines a homomorphism of the additive group ℂ onto the multiplicative group ℂ^×. This epimorphism theorem is basic for everything else; for example, it leads immediately to the realization that there is a uniquely determined positive real number π such that exp z = 1 precisely when z is one of the numbers 2nπi, n ∈ ℤ. This famous constant thereby “occurs naturally among the complex numbers”.

Gauss wrote to Bessel on December 18, 1811: “What should we make of ∫ ϕϰ · dx for ϰ = a + bi? Obviously, if we’re to proceed from clear concepts, we have to assume that ϰ passes, via infinitely small increments (each of the form α + iβ), from that value at which the integral is supposed to be 0, to ϰ = a + bi and that then all the ϕϰ · dx are summed up. In this way the meaning is made precise. But the progression of x values can take place in infinitely many ways: Just as we think of the realm of all real magnitudes as an infinite straight line, so we can envision the realm of all magnitudes, real and imaginary, as an infinite plane wherein every point which is determined by an abscissa α and an ordinate b represents as well the magnitude a + bi.

The era of complex integration begins with CAUCHY. It is consequently condign that his name is associated with practically every major result of this theory. In this chapter the principal Cauchy theorems will be derived in their simplest forms and extensively discussed (sections 1 and 2). We show in section 3 the most important application which is that holomorphic functions may be locally developed into power series. “Ceci marque un des plus grands progrés qui aient jamais été réalisés dans l’Analyse. (This marks one of the greatest advances that have ever been realized in analysis.)” — [Lin], pp. 9,10. As a consequence of the Cauchy-Taylor development of a function we immediately prove (in 3.4) the Riemann continuation theorem, which is indispensable in many subsequent consid-erations. In section 4 we discuss further consequences of the power series theorem. In a closing section we consider the Taylor series of the special functions z cot z, tan z and z / sin z around 0; the coefficients of these series are determined by the so-called Bernoulli numbers. “Le dévelloppement de Taylor rend d’importants services aux mathématiciens. (The Taylor development renders great services to mathematicians.)” — J. Hadamard, 1892

Having led to the Cauchy integral formula and the Cauchy-Taylor representation theorem, the theory of integration in the complex plane will temporarily pass off of center-stage. The power of the two mentioned results has already become clear but this chapter will offer further convincing examples of this power. First off, in section 1 we prove and discuss the Identity Theorem, which makes a statement about the “cohesion among the values taken on by a holomorphic function.” In the second section we illuminate the holomorphy concept from a variety of angles. In the third, the Cauchy estimates are discussed. As applications of them we get, among other things, Liouville’s theorem and, in section 4, the convergence theorems of Weierstrass. The Open Mapping Theorem and the Maximum Principle are proved in section 5.

As soon as Cauchy’s integral formula is available a plethora of themes from classical function theory can be treated directly and independently of each other. This freedom as to choice of themes forces one to impose his own limits; in CarathÉodory ([5], p. viii) we read: “Die größte Schwierigkeit bei der Planung eines Lehrbuches der Funktionentheorie liegt in der Auswahl des Stoffes. Man muß sich von vornherein entschließen, alle Fragen wegzulassen, deren Darstellung zu große Vorbereitungen verlangt. (The greatest difficulty in planning a textbook on function theory lies in the choice of material. You have to decide beforehand to leave aside all questions whose treatment requires too much preparatory development.)”

Functions with singularities are well known from calculus; e.g., the functions are singular at the origin. Although the problem of classifying isolated singularities cannot be satisfactorily solved for functions defined only on ℝ, the situation is quite different in the complex domain. In section 1 we show that isolated singularities of holomorphic functions can be described in a simple way. In section 2, as an application of the classification we study the automorphisms of punctured domains, showing among other things that every automorphism of ℂ is linear.

In 1847 the Berlin mathematician Gotthold Eisenstein (known to students of algebra from his irreducibility criterion) introduced into the theory of the trigonometric functions the series which nowadays are frequently named after him. These Eisenstein series are the simplest examples of normally convergent series of meromorphic functions in ℂ. In this chapter we will first introduce in section 1 the general concepts of compact and normally convergent series meromorphic functions. In section 2 the partial fraction decomposition of the cotangent function will be studied; it is one of the most fruitful series developments in classical analysis. In this section 3 by comparing coefficients from the Taylor series \( \sum\nolimits_{1}^{\infty } {\frac{{2x}}{{{z^{2}} - {v^{2}}}}} {\text{ }}and{\text{ }}\pi {\text{ }}\cot \pi z - \frac{1}{z} \) around 0 we secure the famous Euler identities

In this chapter we discuss two types of series which, after power series, are among the most important series in function theory: Laurent series \( {\sum\nolimits_{{ - \infty }}^{\infty } {{a_{v}}\left( {z - c} \right)} ^{v}} \) and Fourier series \( \sum\nolimits_{{ - \infty }}^{\infty } {{c_{v}}{e^{{2\pi ivz}}}} \). The theory of Laurent series is a theory of power series in annuli; Weierstrass even called Lau rent series power series too (cf. [W₂], p.67). Fourier series are Laurent series around c = 0 with e^2πiz taking over the role of z; their great importance lies in the fact that periodic holomorphic functions can be developed in such series. A particularly important Fourier series is the theta series \( \sum\nolimits_{{ - \infty }}^{\infty } {{e^{{ - {v^{2}}\pi \tau }}}{e^{{2\pi ivz}}}} \), which gave quite a decisive impulse to 19th-century mathematics.

As early as the 18th century many real integrals were evaluated by passing up from the real domain to the complex (passage du réel á l’imaginaire). Especially Euler (Calcul intégrat), Legendre (Exercices de Calcul Intégral) and Laplace made use of this method at a time when the theory of the complex numbers had not yet been rigorously grounded and “all convergence questions still lay under a thick fog.” The attempt to put this procedure on a secure foundation lead Cauchy to the residue calculus.

The residue calculus is eminently suited to evaluating real integrals whose integrands have no known explicit antiderivatives. The basic idea is simple: The real interval of integration is incorporated into a closed path γ in the complex plane and the integrand is then extended into the region bounded by γ. The extension is required to be holomorphic there except for isolated singularities. The integral over γ is then determined from the residue theorem, and the needed residues are computed algebraically. Euler, Laplace and Poisson needed considerable analytic inventiveness to find their integrals. But today it would be more a question of proficiency in the use of the Cauchy formulas. Nevertheless there is no canonical method of finding, for a given integrand and interval of integration, the best path γ in ℂ to use.