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The idea of using invariant geometry to study complex function theory has its foundation in the ideas of Poincaré. Certainly he is credited with the creation of a conformally invariant metric on the unit disk . The uniformization theorem (covered later in this book) may be used to transfer the metric to other planar domains. Later on, Stefan Bergman found a way to define invariant metrics on virtually any domain in any complex manifold. We shall explore his ideas further on in the book.

The Schwarz lemma is one of the simplest results in all of complex function theory. A direct application of the maximum principle, it is merely a statement about the rate of growth of holomorphic functions on the unit disk.

The concept of normal family is an outgrowth of the standard technique for proving the Riemann mapping theorem. Recall that the mapping function is produced as the solution of a certain extremal problem, and showing that that extremal problem actually has a solution is a byproduct of a normal families argument.

The Riemann mapping theorem has been said by some to be the greatest theorem of the nineteenth century. The entire of the theorem is profoundly original, and its proof introduced many new ideas. Certainly normal families and the use of extremal problems in complex analysis are just two of the important techniques that have grown out of studies of the Riemann mapping theorem.

It is a truism that the Riemann mapping theorem allows us to transfer the complex function theory of any simply connected domain (except the plane itself) back to the unit disk, or vice versa. But many of the more delicate questions require something more. If we wish to study behavior of functions at the boundary, or growth or regularity conditions, then we must know something about the boundary behavior of the conformal mapping.

P. Fatou, G. H. Hardy, and F. Riesz were the pioneers in the study of the boundary behavior of holomorphic functions. In 1906, quick on the heels of Lebesgue’s first publications on measure theory, Fatou proved a seminal result about the almost-everywhere boundary limits of bounded, holomorphic functions on the disk. Interestingly, be was able to render the problem as one about convergence of Fourier series, and he solved it in that language.

Certainly every student of complex analysis learns of the Cauchy-Riemann equations These identities, which follow directly from the definition of complex derivative, give an important connection between the real and complex parts of a holomorphic function. Certainly conformality, harmonicity, and many other fundamental ideas are effectively explored by way of the Cauchy—Riemann equations.

Every smoothly bounded domain in the complex plane has a Green’s function. The Green’s function is fundamental to the Poisson integral, the theory of harmonic functions, and to the broad panorama of complex function theory.

Harmonic measure is a device for estimating harmonic functions on a domain. It has become an essential tool in potential theory and in studying the corona problem. It is useful in studying the boundary behavior of conformal mappings, and it tells us a great deal about the boundary behavior of holomorphic functions and solutions of the Dirichlet problem. All these are topics that will be touched on in the present book.

A major theme of analysis in the early twentieth century was the study of convergence of Fourier series. There are two basic types of convergence: pointwise convergence and norm convergence. The study of norm convergence gives rise rather quickly to the study of the Hilbert transform.