Meromorphic functions [reprinted with index]. 9780198535034, 0198535031, 9780198535102, 0198535104

Table of contents :
Acknowledgements
Contents
Introduction
1. The Elementary Theory
1.0. Introduction
1.1. The Poisson-Jensen formula
1.2. The characteristic function
1.3. The first fundamental theorem
1.4. Cartan's identity and convexity theorems
1.5. The Ahlfors-Shimizu characteristic
1.6. An application: average and maximum of n(r, a)
1.7. Orders of growth
1.8. Comparative growth of T(r) and log M(r)
1.9. Representation of a meromorphic function in terms of its zeros and poles
1.10. Behaviour of Weierstrass products
2. Nevanlinna's Second Fundamental Theorem
2.0. Introduction
2.1. The fundamental inequality
2.2. The estimation of S(r)
2.3. Conditions for S(r) to be small
2.4. Nevanlinna's theory of deficient values: the second fundamental theorem
2.5. Some examples
2.6. Deficient functions
2.7. Functions taking the same points
2.8. Fix-points of integral functions
2.9. A theorem of Pólya
3. Distribution of the Values of Meromorphic Functions and their Derivatives
3.0. Introduction
3.1. Milloux theory
3.2. Milloux's basic results
3.3. Exceptional values of meromorphic functions and their derivatives
3.4. Zeros of meromorphic functions and their derivatives. A theorem of Pólya
3.5. The theory of Tumura-Clunie
3.6. Conclusion
4. Further Results about Deficiencies
4.0. Introduction
4.1. An integral function with assigned deficiencies
4.2. On the deficiencies of meromorphic functions of finite order
4.3. The examples of Gol'dberg type
4.4. Some results for functions of non-integral order
4.5. Gol'dberg's theorems on the means of analytic functions
4.6. Deficiencies of functions of genus zero
4.7. Extension of a theorem of Wiman
4.8. Conclusion
5. Ahlfors's Theory of covering Surfaces
5.0. Introduction
5.1. Geometric preliminaries
5.2. Covering surfaces
5.3. The first fundamental theorem
5.4. The topology of surfaces
5.5. A lower bound for the characteristic of a covering surface
5.6. Application to functions meromorphic in a disk
5.7. The principal results of the theory
5.8. Conclusion
6. Functions Meromorphic in the Unit Disk
6.0. Introduction
6.1. The theory of Dufresnoy
6.2. Consequences of Theorem 6.1
6.3. Normal families
6.4. Normal invariant families
6.5. Normal invariant families of regular functions
6.6. Theorems of Schottky, Landau, Bohr and Bloch
6.7. Functions of bounded characteristic
Appendix
Works referred to in the Appendix
Bibliography
Author Index
Index of Terminology