Introduction

Abstract

One of the first theorems we learn as mathematics students is the the Fundamental Theorem of Algebra, which says that a degree d polynomial of one complex variable will have d complex zeros, provided that the zeros are counted with multiplicity. If P(z) is a degree d polynomial, then $\max \{ |P(r{e^{i\theta }})|:0 \leqslant \theta \leqslant 2\pi \}$ grows essentially like r ^d as r → ∞. Therefore, we can rephrase the Fundamental Theorem of Algebra as follows: a non-constant polynomial in one complex variable takes on every finite value an equal number of times counting multiplicity, and that number is determined by the order of growth of the maximum modulus of the polynomial on the circle of radius r centered at the origin as r → ∞. A good way to sum up value distribution theory, otherwise known as Nevanlinna theory, is by saying that the main theorems in value distribution theory are generalizations of the Fundamental Theorem of Algebra to holomorphic and meromorphic functions. The purpose of this book is to describe this theory for meromorphic functions on the complex plane C, or more generally for functions meromorphic in a disc in C.