Extracts from Lars Ahlfors' Complex analysis
The geometric representation of complex numbers
The geometric representation derives its usefulness from the vivid mental pictures associated with a geometric language. If the theorems of synthetic geometry were securely anchored, we could use them to derive results concerning complex numbers. Unfortunately, it must be realized that in elementary courses geometry is not developed with complete rigor in view. For this reason we shall not consider any geometric proofs as conclusive. This limitation does not prevent us from using the geometric language as long as we keep in mind that all proofs must ultimately be reduced to analytic terms. If this is our attitude we are on the other hand relieved from the exigencies of rigor in connection with geometric considerations of a purely descriptive character.Topological concepts
The branch of mathematics which goes under the name of topology is concerned with all questions directly or indirectly related to continuity. The term is traditionally used in a very wide sense and without strict limits. Topological considerations are extremely important for the foundation of the theory of functions, and the first systematic study of topology was motivated by this need.Conformal mappingsIn the present section we are primarily concerned with topological properties of point sets. It must be well understood that the most general properties are the easiest to deal with, because they can be expressed in the simplest logical terms. Many situations which seem intuitively simple are logically quite involved and must be avoided. Sometimes it is therefore necessary to use definitions whose intuitive content is not immediately clear. In such cases we must urge the reader to take a purely formalistic point of view and concentrate on the logical reasoning.
The conformal mapping associated with an analytic function affords an excellent visualization of the properties of the latter; it can well be compared with the visualization of a real function by its graph. It is therefore natural that all questions connected with conformal mapping have received a great deal of attention; progress in this direction has increased our knowledge of analytic functions considerably. In addition, conformal mapping enters naturally in many branches of mathematical physics and in this way accounts for the immediate usefulness of complex function theory. One of the most important problems is to determine the conformal mappings of one region onto another.Complex integration
Many important properties of analytic functions cannot be proved without the use of complex integration. Nobody has been able to prove that the derivative of an analytic function is continuous without resorting to complex integrals or equivalent tools. Even if continuity of the derivative is made part of the definition, it has not been possible to prove the existence of higher derivatives without the use of integration. This failure to eliminate integration from questions which superficially concern only the differential calculus points to deep-rooted differences between complex and real variables.The calculus of residuesAs in the real case we distinguish between definite and indefinite integrals. An indefinite integral is a function whose derivative equals a given analytic function in a region; in many elementary cases indefinite integrals can be found by inversion of known derivation formulas. The definite integrals are taken over differentiable or piecewise differentiable arcs and are not limited to analytic functions. They can be defined by a limit process which mimics the definition of a real definite integral. Actually, we shall prefer to define complex definite integrals in terms of real integrals. This will save us from repeating existence proofs which are essentially the same as in the real case. Naturally, the reader must be thoroughly familiar with the theory of definite integrals of real continuous functions.
The calculus of residues was introduced by Cauchy, the founder of complex integration theory. The calculus of residues provides a very efficient tool for the evaluation of definite integrals. It is particularly important when it is impossible to find the indefinite integral explicitly, but even if the ordinary methods of calculus can be applied the use of residues is frequently a labour saving device. The fact that the calculus of residues yields complex rather than real integrals is no disadvantage, for clearly the evaluation of a complex integral is equivalent to the evaluation of two definite integrals.Analytic functionsThere are, however, some serious limitations, and the method is far from infallible. In the first place, the integrand must be closely connected with some analytic function. This is not very serious, for usually we are only required to integrate elementary functions, and they can all be extended to the complex domain. It is much more serious that the technique of complex integration applies only to closed curves, while a real integral is always extended over an interval. A special device must be used in order to reduce the problem to one which concerns integration over a closed curve. There are a number of ways in which this can be accomplished, but they all apply under rather special circumstances. The technique can be learned at the hand of typical examples, but even complete mastery does not guarantee success.
The central theorem concerning the convergence of analytic functions asserts that the limit of a uniformly convergent sequence of analytic functions is an analytic function. The precise assumptions must be carefully stated, and they should not be too restrictive.Harmonic functionsOne of the most fundamental properties of analytic functions is that they can be represented through convergent power series. Conversely, with trivial exceptions every convergent power series defines an analytic function. Power series are very explicit analytic expressions and as such are extremely maniable. It is therefore not surprising that they turn out to be a powerful tool in the study of analytic functions.
The real and imaginary part of an analytic function are conjugate harmonic functions. Therefore, all theorems on analytic functions are also theorems on pairs of conjugate harmonic functions. However, harmonic functions are important in their own right, and their treatment is not always simplified by the use of complex methods. This is particularly true when the conjugate function is not single-valued and in all questions which concern boundary-value problems.Analytic continuation
In the preceding chapters we have emphasized that all functions must be well defined and, therefore, single-valued. In the case of functions like log z or which are not uniquely determined by their analytic expression, a special effort was needed in order to show that, under favourable circumstances, a single-valued branch can be selected. While this point of view answers the need for logical clarity it does not do justice to the fact that the ambiguity of the logarithm or the square root is an essential characteristic which cannot be ignored. There is thus a clear need for a rigorous theory of multiple-valued functions. We continue to accept the concept of a single-valued analytic function defined in a region as the primary notion in terms of which multiple-valued analytic functions must be defined.Riemann's point of view
Riemann was a strong proponent of the idea that an analytic function can be defined by its singularities and general properties just as well or perhaps even better than through an explicit expression. A trivial example is the determination of a rational function by the singular parts connected with its poles.
JOC/EFR November 2014
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