Ilpo Laine

Risto Korhonen

Kazuya Tohge

This is the first textbook-type presentation of tropical value distribution theory. It provides a detailed introduction of the tropical version of the Nevanlinna theory, describing growth and value distribution analysis of continuous, piecewise linear functions on the real axis. The book also includes applications of this theory to ultra-discrete equations. Three appendices are given to compare the contents of the theory with the classical counterparts in complex analysis. Detailed presentation of the proofs makes the book accessible for lecture courses and independent studies at the graduate and post-doctoral level. Contents:Tropical Polynomials, Rationals and ExponentialsTropical Entire FunctionsOne-dimensional Tropical Nevanlinna TheoryClunie and Mohon'ko Type TheoremsTropical Holomorphic CurvesRepresentations of Tropical Periodic FunctionsApplications to Ultra-Discrete EquationsAppendices:Classical Nevanlinna and Cartan TheoriesIntroduction to Ultra-Discrete Painlevé EquationsSome Operators in Complex Analysis Readership: Graduate students, post-graduates and researchers. Key Features:First monograph type presentation in this topicSome of the material is new in contents, which follows in the presentationThe contributors have been active in value distribution theory, including its tropical variantKeywords:Tropical Mathematics;Value Distribution Theory;Ultra-discrete Equations

Tropical Value Distribution Theory and Ultra-Discrete Equations

Value Distribution Theory and Complex Differential Equations, Joensuu 1994

Seppo Rickman

Tuomas Sorvali

Complex Analysis Joensuu 1987

Summer School in Potential Theory, Joensuu, 1990

The articles in this volume are for the most part research articles related mainly to the theory of quasiconformal and quasiregular mappings, Riemann surfaces and potential theory. They have resulted from talks delivered at the 13th Nevanlinna Colloquium, which was also a celebration of the 80th birthday of Lars V. Ahlfors: hence many articles in this volume reflect his mathematical interests.

Proceedings of the XIIIth Rolf Nevanlinna-Colloquium, Held in Joensuu, Finland, Aug. 10-13, 1987

Rolf Nevanlinna

Olli Martio

Sixteenth Rolf Nevanlinna Colloquium

Kai Liu

Lianzhong Yang

The series is devoted to the publication of monographs and high-level textbooks in mathematics, mathematical methods and their applications. Apart from covering important areas of current interest, a major aim is to make topics of an interdisciplinary nature accessible to the non-specialist. The works in this series are addressed to advanced students and researchers in mathematics and theoretical physics. In addition, it can serve as a guide for lectures and seminars on a graduate level. The series de Gruyter Studies in Mathematics was founded ca. 35 years ago by the late Professor Heinz Bauer and Professor Peter Gabriel with the aim to establish a series of monographs and textbooks of high standard, written by scholars with an international reputation presenting current fields of research in pure and applied mathematics. While the editorial board of the Studies has changed with the years, the aspirations of the Studies are unchanged. In times of rapid growth of mathematical knowledge carefully written monographs and textbooks written by experts are needed more than ever, not least to pave the way for the next generation of mathematicians. In this sense the editorial board and the publisher of the Studies are devoted to continue the Studies as a service to the mathematical community. Please submit any book proposals to Niels Jacob.

Complex Delay-Differential Equations

Valerii I. Gromak

Shun Shimomura

This book is the first comprehensive treatment of Painlevé differential equations in the complex plane. Starting with a rigorous presentation for the meromorphic nature of their solutions, the Nevanlinna theory will be applied to offer a detailed exposition of growth aspects and value distribution of Painlevé transcendents. The subsequent main part of the book is devoted to topics of classical background such as representations and expansions of solutions, solutions of special type like rational and special transcendental solutions, Bäcklund transformations and higher order analogues, treated separately for each of these six equations. The final chapter offers a short overview of applications of Painlevé equations, including an introduction to their discrete counterparts. Due to the present important role of Painlevé equations in physical applications, this monograph should be of interest to researchers in both mathematics and physics and to graduate students interested in mathematical physics and the theory of differential equations.

Painlevé Differential Equations in the Complex Plane

Ilpo Laine

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