The First Main Theorem
As we mentioned in the introduction, the basis for Nevanlinna’s theory are his two “main” theorems. This chapter discusses the first and easier of the two.
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As we mentioned in the introduction, the basis for Nevanlinna's theory are his two "main" theorems. This chapter discusses the first and easier of the two. To understand where we are headed, looking at the concrete example of the transcendental entire function e Z is again helpful. We pointed out that eZ takes on all values other than zero infinitely often. Notice also that by periodicity each of these non-zero values is also taken on with the same asymptotic frequency. Of course e Z never attains the value 0; nor does it attain the value 00. On the other hand, on every large circle centered at the origin, the function e Z spends most of its time close to one of these two omitted values. That is, fixing a small c, the percentage of the circle of radius r where e Z is either smaller than c or larger than 1/c tends to 100% as the radius of the circle tends to infinity. Nevanlinna's First Main Theorem will tell us that this behavior is typical. That is, if a meromorphic function takes on a particular value less often than "expected," then it must compensate for this by spending a lot of time "near" that value. As a consequence, Nevanlinna's First Main Theorem gives an upper bound (in terms of the growth of the function) on how often a meromorphic function can attain any value. This is analogous to the statement that a polynomial of degree d can take on any value at most d times. Note this last statement about polynomials is much easier to prove than the full Fundamental Theorem of Algebra. It is thus no surprise that Nevanlinna's First Main Theorem is much easier than his Second Main Theorem, which says that most values are taken on by a meromorphic function with the maximum asymptotic frequency allowed by the First Main Theorem.
1.1 The Poisson-Jensen Formula We begin this chapter with this brief section which recalls the Poisson Formula for harmonic functions and applies this to the logarithm of the modulus of an analytic function to derive what is known as the "Poisson-Jensen Formula." The material in this first section is usually covered in a first course in complex analysis, and we could have chosen to regard the material in this section as a prerequisite. On the other hand, the Poisson Formula is at the very heart of Nevanlinna's theory of value distribution, and the First Main Theorem of Nevanlinna theory is really nothing W. Cherry et al., Nevanlinna’s Theory of Value Distribution © Springer-Verlag Berlin Heidelberg 2001
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The First Main Theorem
other than the Poisson-Jensen formula dressed up in new notation. Thus, we felt a detailed treatment of this material here would be of value to the reader. We begin with some general notation. We will use C to denote the complex plane. We will use rand R to denote positive real numbers, which will usually be radii of discs, we will always have r < R, and we will also allow R = 00, whenever this makes sense. We will use D(r) and D(R) to denote open discs of radius rand R, respectively, each centered at the origin. When R is allowed to be infinite, D(R) will
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