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## Über dieses Buch

Mathematics is a beautiful subject, and entire functions is its most beautiful branch. Every aspect of mathematics enters into it, from analysis, algebra, and geometry all the way to differential equations and logic. For example, my favorite theorem in all of mathematics is a theorem of R. NevanJinna that two functions, meromorphic in the whole complex plane, that share five values must be identical. For real functions, there is nothing that even remotely corresponds to this. This book is an introduction to the theory of entire and meromorphic functions, with a heavy emphasis on Nevanlinna theory, otherwise known as value-distribution theory. Things included here that occur in no other book (that we are aware of) are the Fourier series method for entire and mero­ morphic functions, a study of integer valued entire functions, the Malliavin­ Rubel extension of Carlson's Theorem (the "sampling theorem"), and the first-order theory of the ring of all entire functions, and a final chapter on Tarski's "High School Algebra Problem," a topic from mathematical logic that connects with entire functions. This book grew out of a set of classroom notes for a course given at the University of Illinois in 1963, but they have been much changed, corrected, expanded, and updated, partially for a similar course at the same place in 1993. My thanks to the many students who prepared notes and have given corrections and comments.

## Inhaltsverzeichnis

### 1. Introduction

Abstract
Mathematics is a beautiful subject, and entire functions is its most beautiful branch. Every aspect of mathematics enters into it, from analysis, algebra, and geometry all the way to differential equations and logic.
Lee A. Rubel, James E. Colliander

### 2. The Riemann-Stieltjes Integral

Abstract
We give here a brief summary of some of the basic facts about the Riemann-Stieltjes integral. Those unfamiliar with the subject are urged to read Chapter 9 of Mathematical Analysis by Apostol [1].
Lee A. Rubel, James E. Colliander

### 3. Jensen’s Theorem and Applications

Abstract
One of our most useful tools is Jensen’s Theorem, which can be used to relate the distribution of zeros of an entire function to its growth. We prove Jensen’s Theorem using the Gauss Mean Value Theorem.
Lee A. Rubel, James E. Colliander

### 4. The First Fundamental Theorem of Nevanlinna Theory

Abstract
Rewriting Jensen’s Theorem, we get
$$\frac{1}{{2\pi }}\int_{{ - \pi }}^{\pi } {\log |f(r{{e}^{{i\theta }}})|} \;d\theta = \log |{{a}_{k}}| + N\left( {r,\frac{1}{f}} \right) - N(r,f),$$
(4.1)
where N is a kind of average number of poles of f.
Lee A. Rubel, James E. Colliander

### 5. Elementary Properties of T(r, f)

Abstract
In this chapter, we present basic properties of the characteristic function.
Lee A. Rubel, James E. Colliander

### 6. The Cartan Formulation of the Characteristic

Abstract
We begin with some remarks on convex functions.
Lee A. Rubel, James E. Colliander

### 7. The Poisson-Jensen Formula

Abstract
The material we present in this chapter is a specialization of some general results of potential theory. Our presentation is in the context of analytic function theory.
Lee A. Rubel, James E. Colliander

### 8. Applications of T(r)

Abstract
Theorem. If f is holomorphic and $$M(r) = \sup [\{ |f(z)|:|z| \leqslant r\}$$, then for any R>r
$$T(r) \leqslant {{\log }^{ + }}M(r) \leqslant \frac{{R + r}}{{R - r}}T(R).$$
Lee A. Rubel, James E. Colliander

### 9. A Lemma of Borel and Some Applications

Abstract
Definition. A set E of real numbers has length ≤ ℓ, written |E|≤ ℓ, means that there is a countable union of intervals [a n ,b n ], a n b n , that contains Eand such that
$$\sum {({{b}_{n}} - {{a}_{n}}) \leqslant \ell .}$$
Lee A. Rubel, James E. Colliander

### 10. The Maximum Term of an Entire Function

Abstract
We will give in this chapter a proof that a suitable entire function can grow as fast as we please.
Lee A. Rubel, James E. Colliander

### 11. Relation Between the Growth of an Entire Function and the Size of Its Taylor Coefficients

Abstract
Let Fbe an entire function and M(r) be its maximum modulus for |z| = r. Suppose λ is a positive continuous increasing function for r ge; 1 such that $$\frac{{\lambda (2r)}}{{\lambda (r)}}$$ is bounded.
Lee A. Rubel, James E. Colliander

### 12. Carleman’s Theorem

Abstract
Let f be holomorphic in Re z ≥ 0 and suppose f has no zeros on z = iy. Choose ρ > 0 so that ρ < (modulus of the smallest zero of f in Re z ≥ 0). Let $$\left\{ {{{z}_{n}} = {{r}_{n}}{{e}^{{i{{\theta }_{n}}}}}} \right\}$$ be the zeros of f in Re z ≥ 0. Define the following:
$$\sum (R) = \sum (R:f) = \sum\limits_{{{{r}_{n}} \leqslant R}} {\left( {\frac{1}{{{{r}_{n}}}} - \frac{{{{r}_{n}}}}{{{{R}^{2}}}}} \right)} \cos {{\theta }_{n}}$$
proper multiplicity of the zeros taken into account);
$$I(R) = I(R:f) = \frac{1}{{2\pi }}\int_{r}^{R} {\left( {\frac{1}{{{{t}^{2}}}} - \frac{1}{{{{R}^{2}}}}} \right)\log \left| {f(it)f( - it)} \right|dt,}$$
where the integral is taken from ir to iR along the imaginary axis; and
$$J(R) = J(R:f) = \frac{1}{{\pi R}}\int_{{ - \pi /2}}^{{\pi /2}} {\log \left| {f({{{\operatorname{Re} }}^{{i\theta }}})} \right|\cos \theta d\theta ,}$$
where the integral is taken along the semicircle of radius R centered at 0. Then
$$\sum (R) = I(R) + J(R) + O(1).$$
Lee A. Rubel, James E. Colliander

### 13. A Fourier Series Method

Abstract
The idea presented in this chapter is the following: If f is a meromorphic function in the complex plane, and if
$${{c}_{k}}(r,f) = \frac{1}{{2\pi }}\int_{{ - \pi }}^{\pi } {(\log |f(r{{e}^{{i\theta }}})|)} {{e}^{{ - ik\theta }}}d\theta$$
is the kth Fourier coefficient of log | f (reiθ)|, then the behavior of f(z) is reflected in the behavior of the sequence {ck(r, f), and vice versa.
Lee A. Rubel, James E. Colliander

### 14. The Miles-Rubel-Taylor Theorem on Quotient Representations of Meromorphic Functions

Abstract
Let f be a meromorphic function. In this chapter we describe the work of Joseph Miles, which completes the work in the last chapter concerning representations of f as the quotient of entire functions with small Nevan-linna characteristic. Miles showed that every set Z of finite λ-density is λ-balanceable. As a consequence of this and the work of Rubel and Taylor in the last chapter, there exist absolute constants A and B such that if f is any meromorphic function in the plane, then f can be expressed as f 1/f 2 where f 1 and f 2are entire functions such that T(r, f i ) ≤ AT(Br, f) for i = 1, 2 and r > 0. It is implicit in the method of proof that for any B > 1 there is a corresponding A for which the desired representation holds for all f. Miles’ proof is ingenious, intricate, and deep. Miles also showed that, in general, B cannot be chosen to be 1 by giving an example of a meromorphic f such that if f = f 1/f 2, where f 1and f 2 are entire, then T(r, f 2) ≠ O(T(r, f)). We do not give this example here.
Lee A. Rubel, James E. Colliander

### 15. Canonical Products

Abstract
We shall suppose that a countable set Z= {zn of complex numbers is given. For convenience, we shall suppose that 0Zand that Z has no finite limit points. More generally, we consider “sets with multiplicity.” This means that some of the z n may be counted multiple times. It is possible to make this notion rigorous, but at the price of clumsier notation. For convenience in this chapter, we shall exclude the null function F(z) = 0 from consideration.
Lee A. Rubel, James E. Colliander

### 16. Formal Power Series

Abstract
We consider the formal power series f (z) = a 0 + a{in1z + a 2 z 2+…, which we usually normalize by a 0 = 0.
Lee A. Rubel, James E. Colliander

### 17. Picard’s Theorem and the Second Fundamental Theorem

Abstract
In this section we shall prove Picard’s Theorem, state the second fundamental theorem of Nevanlinna (leaving the proof for the next section), and derive some of its consequences.
Lee A. Rubel, James E. Colliander

### 18. A Proof of the Second Fundamental Theorem

Abstract
We now begin the proof of the second fundamental theorem of Nevanlinna. We continue to use the convention that all equalities are to be read “modulo O(1).” There are a number of other proofs of the second fundamental theorem, some of them leading to generalizations of it.
Lee A. Rubel, James E. Colliander

### 19. “Two Constant” Theorems and the Phragmén-Lindelöf Theorems

Abstract
The Two Constant Theorem. Suppose that f is holomorphic in $$\mathbb{D} = \{ z:|z| < 1\}$$ and continuous in $$\bar{\mathbb{D}}\backslash \{ 1\}$$. Suppose further that |f| le; N in $$\mathbb{D}$$ and |f| le; M in $$\partial \mathbb{D}\backslash \{ 1\}$$. Then |f| le; M in $$\bar{\mathbb{D}}\backslash \{ 1\}$$.
Lee A. Rubel, James E. Colliander

### 20. The Pólya Representation Theorem

Abstract
The Pólya Representation Theorem plays a central role in the theory of entire functions of exponential-type. We give a somewhat augmented version of this theorem.
Lee A. Rubel, James E. Colliander

### 21. Integer-Valued Entire Functions

Abstract
An integer-valued entire function f is one such that f (n) is an integer for n= 0, 1, 2, • • •. Some examples are
(i)
sinπz

(ii)
2 z

(iii)
any polynomial with integer coefficients.

Lee A. Rubel, James E. Colliander

### 22. On Small Entire Functions of Exponential-Type with Given Zeros

Abstract
This chapter is extracted from a paper of the same name by P. Malliavin and L. A. Rubel [22]. We obtain here a result that considerably generalizes Carlson’s Theorem presented in Chapter 20.
Lee A. Rubel, James E. Colliander

### 23. The First-Order Theory of the Ring of All Entire Functions

Abstract
The material of this chapter is drawn from the paper [3], “First-Order Conformal Invariants.” Let εdenote the ring of all entire functions as an abstract ring. Much information about the theory of entire functions is present in the theory of ε. For example, an entire function f omits the value 7 iff there exists an entire function g such that (f - 7)g= 1.
Lee A. Rubel, James E. Colliander

### 24. Identities of Exponential Functions

Abstract
In this chapter, which is based on [13], we take up some questions prompted by mathematical logic, notably Tarski’s “High School Algebra Problem.” We study identities between certain functions of many variables that are constructed by using the elementary functions of addition x + y, multiplication xy, and one-place exponentiation ex, starting out with all the complex constants and the independent variables z 1,…, z n. We show that every true identity in this class follows from the natural set of 11 axioms of High School Algebra. The major tool in our proofs is the Nevanlinna theory of entire functions of n complex variables, of which we give a brief sketch. It is entirely parallel to the one-variable theory presented in detail earlier in this book. The timid reader can take n = 1, at least for a first reading.
Lee A. Rubel, James E. Colliander

### Backmatter

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