Entire and Meromorphic Functions

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Springer Science & Business Media, 28.02.1996 - 188 Seiten
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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.
 

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Inhalt

Introduction
1
The RiemannStieltjes Integral
3
Jensens Theorem and Applications
6
The First Fundamental Theorem of Nevanlinna Theory
9
Elementary Properties of Trf
12
The Cartan Formulation of the Characteristic
16
The PoissonJensen Formula
20
Applications of Tr
23
Carlemans Theorem
45
A Fourier Series Method
49
The MilesRubelTaylor Theorem on Quotient Representations of Meromorphic Functions
78
Canonical Products
87
Formal Power Series
93
A Proof of the Second Fundamental Theorem
113
IntegerValued Entire Functions
139
The FirstOrder Theory of the Ring of All Entire
158

A Lemma of Borel and Some Applications
26
The Maximum Term of an Entire Function
30
Relation Between the Growth of an Entire Function and the Size of Its Taylor Coefficients
40
Identities of Exponential Functions
175
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