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This book presents developments and new results on complex
differential-difference equations, an area with important and
interesting applications, which also gathers increasing attention.
Key problems, methods, and results related to complex
differential-difference equations are collected to offer an
up-to-date overview of the field.
The Nevanlinna theory of value distribution of meromorphic
functions, one of the milestones of complex analysis during the
last century, was c- ated to extend the classical results
concerning the distribution of of entire functions to the more
general setting of meromorphic functions. Later on, a similar
reasoning has been applied to algebroid functions, subharmonic
functions and meromorphic functions on Riemann surfaces as well as
to - alytic functions of several complex variables, holomorphic and
meromorphic mappings and to the theory of minimal surfaces.
Moreover, several appli- tions of the theory have been exploited,
including complex differential and functional equations, complex
dynamics and Diophantine equations. The main emphasis of this
collection is to direct attention to a number of recently developed
novel ideas and generalizations that relate to the - velopment of
value distribution theory and its applications. In particular, we
mean a recent theory that replaces the conventional consideration
of counting within a disc by an analysis of their geometric
locations. Another such example is presented by the generalizations
of the second main theorem to higher dimensional cases by using the
jet theory. Moreover, s- ilar ideas apparently may be applied to
several related areas as well, such as to partial differential
equations and to differential geometry. Indeed, most of these
applications go back to the problem of analyzing zeros of certain
complex or real functions, meaning in fact to investigate level
sets or level surfaces.
The Nevanlinna theory of value distribution of meromorphic
functions, one of the milestones of complex analysis during the
last century, was c- ated to extend the classical results
concerning the distribution of of entire functions to the more
general setting of meromorphic functions. Later on, a similar
reasoning has been applied to algebroid functions, subharmonic
functions and meromorphic functions on Riemann surfaces as well as
to - alytic functions of several complex variables, holomorphic and
meromorphic mappings and to the theory of minimal surfaces.
Moreover, several appli- tions of the theory have been exploited,
including complex differential and functional equations, complex
dynamics and Diophantine equations. The main emphasis of this
collection is to direct attention to a number of recently developed
novel ideas and generalizations that relate to the - velopment of
value distribution theory and its applications. In particular, we
mean a recent theory that replaces the conventional consideration
of counting within a disc by an analysis of their geometric
locations. Another such example is presented by the generalizations
of the second main theorem to higher dimensional cases by using the
jet theory. Moreover, s- ilar ideas apparently may be applied to
several related areas as well, such as to partial differential
equations and to differential geometry. Indeed, most of these
applications go back to the problem of analyzing zeros of certain
complex or real functions, meaning in fact to investigate level
sets or level surfaces.
This book is the first comprehensive treatment of Painleve
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
Painleve 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, Backlund
transformations and higher order analogues, treated separately for
each of these six equations. The final chapter offers a short
overview of applications of Painleve equations, including an
introduction to their discrete counterparts. Due to the present
important role of Painleve 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.
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. Titles
in planning include Flavia Smarazzo and Alberto Tesei, Measure
Theory: Radon Measures, Young Measures, and Applications to
Parabolic Problems (2019) Elena Cordero and Luigi Rodino,
Time-Frequency Analysis of Operators (2019) Mark M. Meerschaert,
Alla Sikorskii, and Mohsen Zayernouri, Stochastic and Computational
Models for Fractional Calculus, second edition (2020) Mariusz
Lemanczyk, Ergodic Theory: Spectral Theory, Joinings, and Their
Applications (2020) Marco Abate, Holomorphic Dynamics on Hyperbolic
Complex Manifolds (2021) Miroslava Antic, Joeri Van der Veken, and
Luc Vrancken, Differential Geometry of Submanifolds: Submanifolds
of Almost Complex Spaces and Almost Product Spaces (2021) Kai Liu,
Ilpo Laine, and Lianzhong Yang, Complex Differential-Difference
Equations (2021) Rajendra Vasant Gurjar, Kayo Masuda, and Masayoshi
Miyanishi, Affine Space Fibrations (2022)
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.
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Complex Analysis. Joensuu 1978 - Proceedings of the Colloquium on Complex Analysis, Joensuu, Finland, August 24-27, 1978 (English, German, French, Paperback, 1979 ed.)
Ilpo Laine, O.E. Lehto, T. Sorvali
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R1,870
Discovery Miles 18 700
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Ships in 10 - 15 working days
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Romanian Finnish Seminar on Complex Analysis
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.
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