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## Über dieses Buch

This book is devoted to classical and modern achievements in complex analysis. In order to benefit most from it, a first-year university background is sufficient; all other statements and proofs are provided.

We begin with a brief but fairly complete course on the theory of holomorphic, meromorphic, and harmonic functions. We then present a uniformization theory, and discuss a representation of the moduli space of Riemann surfaces of a fixed topological type as a factor space of a contracted space by a discrete group. Next, we consider compact Riemann surfaces and prove the classical theorems of Riemann-Roch, Abel, Weierstrass, etc. We also construct theta functions that are very important for a range of applications.

After that, we turn to modern applications of this theory. First, we build the (important for mathematics and mathematical physics) Kadomtsev-Petviashvili hierarchy and use validated results to arrive at important solutions to these differential equations. We subsequently use the theory of harmonic functions and the theory of differential hierarchies to explicitly construct a conformal mapping that translates an arbitrary contractible domain into a standard disk – a classical problem that has important applications in hydrodynamics, gas dynamics, etc.

The book is based on numerous lecture courses given by the author at the Independent University of Moscow and at the Mathematics Department of the Higher School of Economics.

## Inhaltsverzeichnis

### Chapter 1. Holomorphic Functions

Abstract
By a domain we mean a connected open subset of the complex plane. The correspondence (x, y) ↔ z = x + iy between the real plane $$\mathbb {R}^2$$ and the complex plane $$\mathbb {C}$$ allows one to regard a complex-valued function of a complex variable as
• a map from a domain $$D \subset \mathbb {C}$$ in the complex plane to the complex plane $$\mathbb {C}$$ (notation: w = f(z));
• a map from a domain $$D \subset \mathbb {R}^2$$ in the real plane to the complex plane $$\mathbb {C}$$ (notation: w = f(x, y));
• a map from a domain $$D \subset \mathbb {R}^2$$ in the real plane to the real plane $$\mathbb {R}^2$$ (notation: (u, v) = f(x, y), u = u(x, y), v = v(x, y)).
In what follows, we will often switch between these interpretations.
Sergey M. Natanzon

### Chapter 2. Meromorphic Functions

Abstract
Now we turn to studying the properties of functions holomorphic in non-simply connected domains.
Sergey M. Natanzon

### Chapter 3. Riemann Mapping Theorem

Abstract
A family $${\frak F}$$ of functions is said to be uniformly bounded inside a domain D if for every compact set K ⊂ D there exists a constant M = M(K) such that |f(z)|≤ M for all $$f \in {\frak F}$$, z ∈ K.
Sergey M. Natanzon

### Chapter 4. Harmonic Functions

Abstract
As before, we identify the real plane $$\mathbb {R} \times \mathbb {R} = \{ (x,y) \}$$ with the complex plane $${\mathbb {C} = \{ z \}}$$ setting z = x + iy. Recall that an open connected subset $$D \subset \mathbb {C} = \mathbb {R}^2$$ is called a domain.
Sergey M. Natanzon

### Chapter 5. Riemann Surfaces and Their Modules

Abstract
A Riemann surface is a one-dimensional complex manifold. A Riemann surface is defined as an equivalence class of atlases of charts on a surface with biholomorphic transition maps.
Sergey M. Natanzon

### Chapter 6. Compact Riemann Surfaces

Abstract
In this section, we will consider only compact Riemann surfaces of genus g, i.e., surfaces of type (g, 0, 0). Such a surface is homeomorphic to a sphere with g holes in which every boundary contour is glued to the boundary contour of a torus with a hole. Recall that a complex structure on a surface is defined by a holomorphic atlas of local charts. A map between surfaces is said to be holomorphic if it is holomorphic in every local chart.
Sergey M. Natanzon

### Chapter 7. The Riemann–Roch Theorem and Theta Functions

Abstract
A finite formal linear combination
$$\displaystyle D = \sum \limits _{i=1}^k n_i p_i$$
of points p i ∈ P of a Riemann surface P with integer coefficients $$n_i\in \mathbb Z$$ is called a divisor on P. The set of divisors is a module over the ring of integers $$\mathbb Z$$. The zero element of this module is $$D = \varnothing$$, a sum with no terms.
Sergey M. Natanzon

### Chapter 8. Integrable Systems

Abstract
In what follows, we use the binomial coefficients $${p\choose t}$$ given by $${p\choose t}=\frac {p!}{t!(p-t)!}$$ for p ≥ t ≥ 0 and $${p\choose t}= 0$$ otherwise, with the standard convention that 0! = 1.
Sergey M. Natanzon

### Chapter 9. Formula for a Conformal Map from an Arbitrary Domain onto Disk

Abstract
Conformal maps are essential in a broad range of applied problems (aeromechanics and hydromechanics, oil production, etc.). So, we would like to refine Riemann’s theorem on the existence of a conformal map from a simply connected domain D to the unit disk Λ by providing an explicit construction of such a map.
Sergey M. Natanzon

### Backmatter

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