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This is the first book on the theory of multiple zeta values since its birth around 1994. Readers will find that the shuffle products of multiple zeta values are applied to complicated counting problems in combinatorics, producing numerous interesting identities that are ready to be used. This will provide a powerful tool to deal with problems in multiple zeta values, both in evaluations and shuffle relations. The volume will benefit graduate students doing research in number theory.

Functions, Zeta. --- Algebraic functions. --- Functions, Algebraic --- Functions --- Zeta functions --- Functions of complex variables. --- Combinatorial analysis.

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This book is intended for a graduate course in complex analysis, where the main focus is the theory of complex-valued functions of a single complex variable. This theory is a prerequisite for the study of many areas of mathematics, including the theory of several finitely and infinitely many complex variables, hyperbolic geometry, two- and three-manifolds, and number theory. Complex analysis has connections and applications to many other subjects in mathematics and to other sciences. Thus this material will also be of interest to computer scientists, physicists, and engineers. The book covers most, if not all, of the material contained in Lipman Bers’s courses on first year complex analysis. In addition, topics of current interest, such as zeros of holomorphic functions and the connection between hyperbolic geometry and complex analysis, are explored. In addition to many new exercises, this second edition introduces a variety of new and interesting topics. New features include a section on Bers's theorem on isomorphisms between rings of holomorphic functions on plane domains; necessary and sufficient conditions for the existence of a bounded analytic function on the disc with prescribed zeros; sections on subharmonic functions and Perron's principle; and a section on the ring of holomorphic functions on a plane domain.  There are three new appendices: the first is a contribution by Ranjan Roy on the history of complex analysis, the second contains background material on exterior differential calculus, and the third appendix includes an alternate approach to the Cauchy theory.

Functions of complex variables. --- Mathematical analysis. --- Mathematics. --- Functions of complex variables --- Mathematical analysis --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- 517.1 Mathematical analysis --- Complex variables --- Functions of a Complex Variable. --- Elliptic functions --- Functions of real variables

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Evolutionary equations are studied in abstract Banach spaces and in spaces of bounded number sequences. For linear and nonlinear difference equations, which are defined on finite-dimensional and infinite-dimensional tori, the problem of reducibility is solved, in particular, in neighborhoods of their invariant sets, and the basics for a theory of invariant tori and bounded semi-invariant manifolds are established. Also considered are the questions on existence and approximate construction of periodic solutions for difference equations in infinite-dimensional spaces and the problem of extendibi

Banach spaces. --- Evolution equations. --- Evolutionary equations --- Equations, Evolution --- Equations of evolution --- Differential equations --- Functions of complex variables --- Generalized spaces --- Topology --- Differential equations, Nonlinear.

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Derivatives of Inner Functions was inspired by a conference held at the Fields Institute in 2011 entitled "Blaschke Products and Their Applications." Inner functions form an important subclass of bounded analytic functions. Since they have unimodular boundary values, they appear in many extremal problems of complex analysis. They have been extensively studied since the early twentieth century and the literature on this topic is vast. This book is devoted to a concise study of derivatives of inner functions and is confined to treating the integral means of derivatives and presenting a comprehensive list of results on Hardy and Bergman means. This self-contained monograph allows researchers to get acquainted with the essentials of inner functions, rendering this theory accessible to graduate students while providing the reader with rapid access to the frontiers of research in this field.

Fluid mechanics. --- Functional analysis --- Functions of complex variables --- Differential equations, Partial --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Functions of complex variables. --- Functional analysis. --- Functional calculus --- Complex variables --- Mathematics. --- Functions of a Complex Variable. --- Functional Analysis. --- Several Complex Variables and Analytic Spaces. --- Calculus of variations --- Functional equations --- Integral equations --- Elliptic functions --- Functions of real variables --- Differential equations, partial. --- Partial differential equations

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Hidden Harmony—Geometric Fantasies describes the history of complex function theory from its origins to 1914, when the essential features of the modern theory were in place. It is the first history of mathematics devoted to complex function theory, and it draws on a wide range of published and unpublished sources. In addition to an extensive and detailed coverage of the three founders of the subject—Cauchy, Riemann, and Weierstrass—it looks at the contributions of great mathematicians from d’Alembert to Poincaré, and Laplace to Weyl. Select chapters examine the rise and importance of elliptic function theory, differential equations in the complex domain, geometric function theory, and the early years of complex function theory in several variables. Unique emphasis has been placed on the creation of a textbook tradition in complex analysis by considering some seventy textbooks in nine different languages. This book is not a mere sequence of disembodied results and theories, but offers a comprehensive picture of the broad cultural and social context in which the main players lived and worked by paying attention to the rise of mathematical schools and of contrasting national traditions. This work is unrivaled for its breadth and depth, both in the core theory and its implications for other fields of mathematics. It is a major resource for professional mathematicians as well as advanced undergraduate and graduate students and anyone studying complex function theory.

Functional analysis. --- Functions of complex variables. --- Functions, Quaternion. --- Number theory. --- Functions of complex variables --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Mathematical analysis. --- 517.1 Mathematical analysis --- Mathematical analysis --- Complex variables --- Mathematics. --- History. --- Functional Analysis. --- History of Mathematical Sciences. --- Functions of a Complex Variable. --- Number Theory. --- Elliptic functions --- Functions of real variables --- Number study --- Numbers, Theory of --- Algebra --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Annals --- Auxiliary sciences of history --- Math --- Science --- Functional analysis --- Number Theory