*Complex analysis*. We choose them to illustrate Ahlfors' style. These are from the First Edition published in 1953:

**The geometric representation of complex numbers**

**Topological concepts**

In 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.

**Conformal mappings**

**Complex integration**

As 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**

There 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.

**Analytic functions**

One 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.

**Harmonic functions**

**Analytic continuation**

**Riemann's point of view**