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Complex Analysis and Special Functions

Autor: Valery Serov , Markus Harju

Número de Páginas: 424

The first two parts of this book focus on developing standard analysis concepts in the extended complex plane. We cover differentiation and integration of functions of one complex variable. Famous Cauchy formulas are established and applied in the frame of residue theory. Taylor series is used to investigate analytic functions, and they are connected to harmonic functions. Laurent series theory is developed. The third part of the book finds applications of the earlier chapter in conformal mappings and the Laplace transform. Special functions solving ordinary differential equations are studied extensively, along with their asymptotic behavior. A highlight of the book is the elliptic function of Weierstrass and Jacobi. Finally, we present Laplace’s method, which is applied to find large arguments asymptotic of some special functions. The book is filled with examples, exercises, and problems of varying degrees of difficulty. This makes it useful to all students in mathematics, physics, and related fields.

SPECIAL FUNCTIONS AND COMPLEX VARIABLES (ENGINEERING MATHEMATICS III)

Autor: Bathul, Shahnaz

Número de Páginas: 586

This thoroughly revised book, now in its third edition, continues to discuss two important topics—special functions and complex variables. Chapters have been rearranged keeping in view the current syllabi of the universities. The book analyzes special functions, Legendre’s equation and function, and Bessel’s function. It explains how to solve Cauchy equations, differential equation with variable coefficients and Frobenius of solving differential equation at a regular singular point. Besides, the text also explains the notions of limit, continuity and differentiability by giving a thorough grounding on analytic functions and their relations with harmonic functions. In addition, the book introduces the exponential function of a complex variable, and with the help of this function, defines trigonometric and hyperbolic functions and explains their properties. While discussing different mathematical concepts, the book discusses a number of theorems such as Cauchy’s integral theorem for the integration of a complex variable, Taylor’s theorem for the analysis of complex power series, the residue theorem for evaluation of residues, the argument principle and Rouche’s theorem...

COMPLEX VARIABLES AND SPECIAL FUNCTIONS

Autor: Baidyanath Patra

Número de Páginas: 494

Author's aim is to make the readers easily understand the theory of complex variables. He explains this subject matter from a rudimentary to advanced level in a very simple manner. Organized in two parts, this book explains exact definitions of different terms used by supplying worked-out examples wherever found necessary. A large number of examples have been solved in the book to acquaint the readers with different techniques. Furthermore, a large number of problems have been supplied with answers at the end of each chapter. The first part of the book (Chapters 1 through 11) containing analysis of complex variables will be useful for the undergraduate students of engineering and science. The second part of the book (Chapters 12 through 20) is written in complex domain and is targeted towards advanced level readers who are either pursuing postgraduate studies in Mathematics or research in Applied Mathematics. The first part is prerequisite for this section of the book.

Complex Analysis and Special Functions

Autor: Yasmine Abouelseoud

Número de Páginas: 0

These Mathematics lecture notes aim to help the student 1) Understand the concept of complex functions as mappings 2) Learn new transforms and their applications 3) Learn about special functions and their applications

SPECIAL FUNCTIONS AND COMPLEX VARIABLES

Autor: Shahnaz Bathul

Número de Páginas: 534

This well-received book, which is a new edition of Textbook of Engineering Mathematics: Special Functions and Complex Variables by the same author, continues to discuss two important topics—special functions and complex variables. It analyzes special functions such as gamma and beta functions, Legendre’s equation and function, and Bessel’s function. Besides, the text explains the notions of limit, continuity and differentiability by giving a thorough grounding on analytic functions and their relations with harmonic functions. In addition, the book introduces the exponential function of a complex variable and, with the help of this function, defines the trigonometric and hyperbolic functions and explains their properties. While discussing different mathematical concepts, the book analyzes a number of theorems such as Cauchy’s integral theorem for the integration of a complex variable, Taylor’s theorem for the analysis of complex power series, the residue theorem for evaluation of residues, besides the argument principle and Rouche’s theorem for the determination of the number of zeros of complex polynomials. Finally, the book gives a thorough exposition of conformal...

Complex Analysis and Special Functions with Mathematical Software Tools

Autor: A. Swaminathan

Número de Páginas: 550

This text emphasizes the special functions that are used in complex analysis. Starting with the algebraic system of complex numbers, it offers an entry-level course on complex analysis of one variable. It presents the study of analytic functions, conformal mapping, analysis of singularities, and the computation of various integrals. The final three chapters introduce more advanced topics and applications. The book provides examples of applications to various physical problems and explains how to use Mathematica®, MapleTM, and MATLAB®.

Numerical Methods for Special Functions

Autor: Amparo Gil , Javier Segura , Nico M. Temme

Número de Páginas: 419

Special functions arise in many problems of pure and applied mathematics, mathematical statistics, physics, and engineering. This book provides an up-to-date overview of numerical methods for computing special functions and discusses when to use these methods depending on the function and the range of parameters. Not only are standard and simple parameter domains considered, but methods valid for large and complex parameters are described as well. The first part of the book (basic methods) covers convergent and divergent series, Chebyshev expansions, numerical quadrature, and recurrence relations. Its focus is on the computation of special functions; however, it is suitable for general numerical courses. Pseudoalgorithms are given to help students write their own algorithms. In addition to these basic tools, the authors discuss other useful and efficient methods, such as methods for computing zeros of special functions, uniform asymptotic expansions, Padé approximations, and sequence transformations. The book also provides specific algorithms for computing several special functions (like Airy functions and parabolic cylinder functions, among others).

Complex Analysis with Applications to Flows and Fields

Autor: Luis Manuel Braga Da Costa Campos

Número de Páginas: 1010

Complex Analysis with Applications to Flows and Fields presents the theory of functions of a complex variable, from the complex plane to the calculus of residues to power series to conformal mapping. The book explores numerous physical and engineering applications concerning potential flows, the gravity field, electro- and magnetostatics, steady he

Complex Functions: An Introduction to Complex Analysis

Autor: Lexa N. Palmer

Número de Páginas: 175

Discover the elegant and powerful world of complex analysis in this comprehensive introduction to one of mathematics' most beautiful subjects. "Complex Functions: An Introduction to Complex Analysis" bridges the gap between introductory calculus and advanced mathematical theory, revealing how the simple addition of the imaginary unit transforms mathematics into something extraordinary. Complex analysis stands as a cornerstone of modern mathematics, physics, and engineering, offering tools of remarkable power and elegance. Whether you're a mathematics student seeking deeper understanding, a physicist requiring analytical techniques, or an engineer solving practical problems, this book provides the foundation you need to master this essential field. Written with clarity and precision, this text balances theoretical rigor with intuitive explanations, making abstract concepts accessible without sacrificing mathematical depth. From the foundations of complex numbers to the frontiers of modern research, this book guides you through the fascinating landscape of complex functions with carefully crafted examples and applications. What you will find in this book: A systematic development of ...

Special Functions & Their Applications

Autor: N. N. Lebedev

Número de Páginas: 340

Famous Russian work discusses the application of cylinder functions and spherical harmonics; gamma function; probability integral and related functions; Airy functions; hyper-geometric functions; more. Translated by Richard Silverman.

Special Functions and Orthogonal Polynomials

Autor: Refaat El Attar

Número de Páginas: 312

(308 Pages). This book is written to provide an easy to follow study on the subject of Special Functions and Orthogonal Polynomials. It is written in such a way that it can be used as a self study text. Basic knowledge of calculus and differential equations is needed. The book is intended to help students in engineering, physics and applied sciences understand various aspects of Special Functions and Orthogonal Polynomials that very often occur in engineering, physics, mathematics and applied sciences. The book is organized in chapters that are in a sense self contained. Chapter 1 deals with series solutions of Differential Equations. Gamma and Beta functions are studied in Chapter 2 together with other functions that are defined by integrals. Legendre Polynomials and Functions are studied in Chapter 3. Chapters 4 and 5 deal with Hermite, Laguerre and other Orthogonal Polynomials. A detailed treatise of Bessel Function in given in Chapter 6.

Análisis funcional

Autor: Walter Rudin

Número de Páginas: 410

Este es un texto que contiene una exposición axiomática de la teoría general de espacios vectoriales topológicos y trata con profundidad algunos aspectos de la teoría y, asimismo, muestra varios ejemplos interesantes de aplicación a otras ramas de la Matemática.

Complex Analysis

Autor: John Stalker

Número de Páginas: 240

In this concise introduction to the classical theory of one complex variable the content is driven by techniques and examples, rather than definitions and theorems.

Handbook of Complex Analysis

Autor: Reiner Kuhnau

Número de Páginas: 876

Geometric Function Theory is that part of Complex Analysis which covers the theory of conformal and quasiconformal mappings. Beginning with the classical Riemann mapping theorem, there is a lot of existence theorems for canonical conformal mappings. On the other side there is an extensive theory of qualitative properties of conformal and quasiconformal mappings, concerning mainly a prior estimates, so called distortion theorems (including the Bieberbach conjecture with the proof of the Branges). Here a starting point was the classical Scharz lemma, and then Koebe's distortion theorem. There are several connections to mathematical physics, because of the relations to potential theory (in the plane). The Handbook of Geometric Function Theory contains also an article about constructive methods and further a Bibliography including applications eg: to electroxtatic problems, heat conduction, potential flows (in the plane). · A collection of independent survey articles in the field of GeometricFunction Theory · Existence theorems and qualitative properties of conformal and quasiconformal mappings · A bibliography, including many hints to applications in electrostatics, heat...

Some Topics in Complex Analysis and Its Apps. in Special Functions

Autor: Ahmed Adly Mahmoud Abdelhafez , Mohamed S. Metwally , Kamel A. M. Sayyed

Número de Páginas: 80

The subject of the special functions of the single complex variable and several complex variables occupies a great attention in almost all disciplines Mathematics, Physics and Engineering. The present work is devoted to study this subject, taking it into account as one of several theses, which presented and still being presented by Specialists in complex analysis. The main aim of this thesis de ning and studying of some special functions which contain two complex variables such as Bessel matrix function, Tricomi matrix function, Horn matrix function and Struve matrix function and also providing special inequalities of Bessel functions and modi ed Bessel functions and Tricomi functions of two scalar index of two complex variables and of their ratios."

Orthogonal Polynomials and Special Functions

Autor: Erik Koelink , Walter Van Assche

Número de Páginas: 259

The set of lectures from the Summer School held in Leuven in 2002 provide an up-to-date account of recent developments in orthogonal polynomials and special functions, in particular for algorithms for computer algebra packages, 3nj-symbols in representation theory of Lie groups, enumeration, multivariable special functions and Dunkl operators, asymptotics via the Riemann-Hilbert method, exponential asymptotics and the Stokes phenomenon. Thenbsp;volume aims at graduate students and post-docs working in the field of orthogonal polynomials and special functions, and in related fields interacting with orthogonal polynomials, such as combinatorics, computer algebra, asymptotics, representation theory, harmonic analysis, differential equations, physics. The lectures are self-contained requiring onlynbsp;a basic knowledge of analysis and algebra, and each includes many exercises.

Special Functions

Autor: Refaat El Attar

Número de Páginas: 311

(Hardcover). This book is written to provide an easy to follow study on the subject of Special Functions and Orthogonal Polynomials. It is written in such a way that it can be used as a self study text. Basic knowledge of calculus and differential equations is needed. The book is intended to help students in engineering, physics and applied sciences understand various aspects of Special Functions and Orthogonal Polynomials that very often occur in engineering, physics, mathematics and applied sciences. The book is organized in chapters that are in a sense self contained. Chapter 1 deals with series solutions of Differential Equations. Gamma and Beta functions are studied in Chapter 2 together with other functions that are defined by integrals. Legendre Polynomials and Functions are studied in Chapter 3. Chapters 4 and 5 deal with Hermite, Laguerre and other Orthogonal Polynomials. A detailed treatise of Bessel Function in given in Chapter 6.

Análisis real y complejo

Autor: Walter Rudin

Número de Páginas: 397

Complex Analysis – Methods, Trends, and Applications

Autor: Eberhard Lanckau , Wolfgang Tutschke

Número de Páginas: 400

No detailed description available for "Complex Analysis – Methods, Trends, and Applications".

Complex Analysis

Autor: Shashank Tiwari

Número de Páginas: 327

"Complex Analysis: Advanced Concepts" delves into the intricate world of complex numbers and functions, offering a thorough exploration of their properties and applications. The book begins with a detailed examination of basic concepts, covering arithmetic operations, geometric interpretations, and the fundamental theorem of algebra. It then progresses to advanced topics such as complex functions, differentiation, integration, and series. One of the book's notable strengths lies in its clear and concise explanations, accompanied by numerous examples and exercises to reinforce understanding. Readers are guided through theorems and proofs, gaining insight into the elegance and power of complex analysis. The book also highlights the relevance of complex analysis in various fields, including physics, engineering, and economics. Applications such as potential theory, fluid dynamics, and signal processing are explored, demonstrating the subject's practical significance. Whether used as a textbook for students or a reference for professionals, "Complex Analysis: Advanced Concepts" offers a valuable resource for mastering the intricacies of this essential branch of mathematics. Its...

Special Functions and Analysis of Differential Equations

Autor: Praveen Agarwal , Ravi P Agarwal , Michael Ruzhansky

Número de Páginas: 371

Differential Equations are very important tools in Mathematical Analysis. They are widely found in mathematics itself and in its applications to statistics, computing, electrical circuit analysis, dynamical systems, economics, biology, and so on. Recently there has been an increasing interest in and widely-extended use of differential equations and systems of fractional order (that is, of arbitrary order) as better models of phenomena in various physics, engineering, automatization, biology and biomedicine, chemistry, earth science, economics, nature, and so on. Now, new unified presentation and extensive development of special functions associated with fractional calculus are necessary tools, being related to the theory of differentiation and integration of arbitrary order (i.e., fractional calculus) and to the fractional order (or multi-order) differential and integral equations. This book provides learners with the opportunity to develop an understanding of advancements of special functions and the skills needed to apply advanced mathematical techniques to solve complex differential equations and Partial Differential Equations (PDEs). Subject matters should be strongly related...

The Complex Web: Unveiling the Labyrinth of Complex Analysis

Autor: Pasquale De Marco

Número de Páginas: 161

In "The Complex Web: Unveiling the Labyrinth of Complex Analysis," embark on an intellectual journey into the captivating world of complex analysis, a branch of mathematics that unlocks the mysteries of functions of complex variables. Within these pages, you'll find a comprehensive exploration of this intricate field, unraveling its fundamental concepts, groundbreaking theorems, and diverse applications. Delve into the rich history of complex analysis, tracing its evolution from its early origins to its current state. Discover the contributions of brilliant mathematicians who shaped the field, revolutionizing our understanding of complex numbers and their applications. Witness the birth of groundbreaking ideas, the resolution of long-standing mathematical conundrums, and the emergence of powerful techniques that have transformed the landscape of mathematics. Explore the intricate tapestry of complex functions, uncovering their unique properties and behaviors. Master the art of complex differentiation and integration, unlocking the secrets of complex derivatives and integrals. Delve into the realm of complex power series, discovering their remarkable convergence properties and...

Introduction to Complex Analysis

Autor: Michael E. Taylor

Número de Páginas: 497

In this text, the reader will learn that all the basic functions that arise in calculus—such as powers and fractional powers, exponentials and logs, trigonometric functions and their inverses, as well as many new functions that the reader will meet—are naturally defined for complex arguments. Furthermore, this expanded setting leads to a much richer understanding of such functions than one could glean by merely considering them in the real domain. For example, understanding the exponential function in the complex domain via its differential equation provides a clean path to Euler's formula and hence to a self-contained treatment of the trigonometric functions. Complex analysis, developed in partnership with Fourier analysis, differential equations, and geometrical techniques, leads to the development of a cornucopia of functions of use in number theory, wave motion, conformal mapping, and other mathematical phenomena, which the reader can learn about from material presented here. This book could serve for either a one-semester course or a two-semester course in complex analysis for beginning graduate students or for well-prepared undergraduates whose background includes...

Using the Mathematics Literature

Autor: Kristine K. Fowler

Número de Páginas: 412

This reference serves as a reader-friendly guide to every basic tool and skill required in the mathematical library and helps mathematicians find resources in any format in the mathematics literature. It lists a wide range of standard texts, journals, review articles, newsgroups, and Internet and database tools for every major subfield in mathematics and details methods of access to primary literature sources of new research, applications, results, and techniques. Using the Mathematics Literature is the most comprehensive and up-to-date resource on mathematics literature in both print and electronic formats, presenting time-saving strategies for retrieval of the latest information.

Lecture Notes On Applied Analysis

Autor: Roderick S C Wong

Número de Páginas: 303

There are several subjects in analysis that are frequently used in applied mathematics, theoretical physics and engineering sciences, such as complex variable, ordinary differential equations, special functions, asymptotic methods, integral transforms and distribution theory. However, for graduate students or upper-level undergraduate students who are not going to specialize in these areas, there is no need for them to study these subjects in great depth. Instead, it would probably be more beneficial for them to have an introduction to these topics so that when the need arises, they know what approach to take. With this in mind, this set of lecture notes has been written for a one-semester course. Sufficient details have also been included to make it sufficiently adaptable for self-study. There are in total six chapters with each covering only a few topics. Furthermore, the chapters are all self-contained. The prerequisites for the readers of this book are advanced calculus, a first course in ordinary differential equations and elementary complex variable.

Asymptotics and Special Functions

Autor: F. W. J. Olver

Número de Páginas: 589

Asymptotics and Special Functions provides a comprehensive introduction to two important topics in classical analysis: asymptotics and special functions. The integrals of a real variable and contour integrals are discussed, along with the Liouville-Green approximation and connection formulas for solutions of differential equations. Differential equations with regular singularities are also considered, with emphasis on hypergeometric and Legendre functions. Comprised of 14 chapters, this volume begins with an introduction to the basic concepts and definitions of asymptotic analysis and special functions, followed by a discussion on asymptotic theories of definite integrals containing a parameter. Contour integrals as well as integrals of a real variable are described. Subsequent chapters deal with the analytic theory of ordinary differential equations; differential equations with regular and irregular singularities; sums and sequences; and connection formulas for solutions of differential equations. The book concludes with an evaluation of methods used in estimating (as opposed to bounding) errors in asymptotic approximations and expansions. This monograph is intended for graduate...

Library of Congress Subject Headings

Autor: Library Of Congress , Library Of Congress. Office For Subject Cataloging Policy

Número de Páginas: 1536

Complex Analysis and Special Topics in Harmonic Analysis

Autor: Carlos A. Berenstein , Roger Gay

Número de Páginas: 491

A companion volume to the text "Complex Variables: An Introduction" by the same authors, this book further develops the theory, continuing to emphasize the role that the Cauchy-Riemann equation plays in modern complex analysis. Topics considered include: Boundary values of holomorphic functions in the sense of distributions; interpolation problems and ideal theory in algebras of entire functions with growth conditions; exponential polynomials; the G transform and the unifying role it plays in complex analysis and transcendental number theory; summation methods; and the theorem of L. Schwarz concerning the solutions of a homogeneous convolution equation on the real line and its applications in harmonic function theory.

Essential Mathematical Methods for Physicists, ISE

Autor: Hans J. Weber , George B. Arfken

Número de Páginas: 960

This new adaptation of Arfken and Weber's best-selling Mathematical Methods for Physicists, fifth edition, is the most modern collection of mathematical principles for solving physics problems.

Applications of Complex Variables

Autor: Foluso Ladeinde

Número de Páginas: 606

The subject of applied complex variables is so fundamental that most of the other topics in advanced engineering mathematics (AEM) depend on it. The present book contains complete coverage of the subject, summarizing the more elementary aspects that you find in most AEM textbooks and delving into the more specialized topics that are less commonplace. The book represents a one-stop reference for complex variables in engineering analysis. The applications of conformal mapping in this book are significantly more extensive than in other AEM textbooks. The treatments of complex integral transforms enable a much larger class of functions that can be transformed, resulting in an expanded use of complex-transform techniques in engineering analysis. The inclusion of the asymptotics of complex integrals enables the analysis of models with irregular singular points. The book, which has more than 300 illustrations, is generous with realistic example problems.

L2 Approaches in Several Complex Variables

Autor: Takeo Ohsawa

Número de Páginas: 202

The purpose of this monograph is to present the current status of a rapidly developing part of several complex variables, motivated by the applicability of effective results to algebraic geometry and differential geometry. Highlighted are the new precise results on the L2 extension of holomorphic functions. In Chapter 1, the classical questions of several complex variables motivating the development of this field are reviewed after necessary preparations from the basic notions of those variables and of complex manifolds such as holomorphic functions, pseudoconvexity, differential forms, and cohomology. In Chapter 2, the L2 method of solving the d-bar equation is presented emphasizing its differential geometric aspect. In Chapter 3, a refinement of the Oka–Cartan theory is given by this method. The L2 extension theorem with an optimal constant is included, obtained recently by Z. Błocki and by Q.-A. Guan and X.-Y. Zhou separately. In Chapter 4, various results on the Bergman kernel are presented, including recent works of Maitani–Yamaguchi, Berndtsson, and Guan–Zhou. Most of these results are obtained by the L2 method. In the last chapter, rather specific results are...

Complex Analysis

Autor: Teodor Bulboacǎ , Santosh B. Joshi , Pranay Goswami

Número de Páginas: 594

This book is an in-depth and modern presentation of important classical results in complex analysis and is suitable for a first course on the topic, as taught by the authors at several universities. The level of difficulty of the material increases gradually from chapter to chapter, and each chapter contains many exercises with solutions and applications of the results, with the particular goal of showcasing a variety of solution techniques.

Library of Congress Subject Headings

Autor: Library Of Congress. Cataloging Policy And Support Office

Número de Páginas: 1588

Special Functions and Orthogonal Polynomials

Autor: Richard Beals , Roderick Wong

Número de Páginas: 489

A comprehensive graduate-level introduction to classical and contemporary aspects of special functions.

Complex Analysis with MATHEMATICA®

Autor: William T. Shaw

Número de Páginas: 6

This book presents a way of learning complex analysis, using Mathematica. Includes CD with electronic version of the book.

A Complex Analysis Problem Book

Autor: Daniel Alpay

Número de Páginas: 592

This second edition presents a collection of exercises on the theory of analytic functions, including completed and detailed solutions. It introduces students to various applications and aspects of the theory of analytic functions not always touched on in a first course, while also addressing topics of interest to electrical engineering students (e.g., the realization of rational functions and its connections to the theory of linear systems and state space representations of such systems). It provides examples of important Hilbert spaces of analytic functions (in particular the Hardy space and the Fock space), and also includes a section reviewing essential aspects of topology, functional analysis and Lebesgue integration. Benefits of the 2nd edition Rational functions are now covered in a separate chapter. Further, the section on conformal mappings has been expanded.

Complex Variables and Analytic Functions

Autor: Bengt Fornberg , Cécile Piret

Número de Páginas: 372

At almost all academic institutions worldwide, complex variables and analytic functions are utilized in courses on applied mathematics, physics, engineering, and other related subjects. For most students, formulas alone do not provide a sufficient introduction to this widely taught material, yet illustrations of functions are sparse in current books on the topic. This is the first primary introductory textbook on complex variables and analytic functions to make extensive use of functional illustrations. Aiming to reach undergraduate students entering the world of complex variables and analytic functions, this book utilizes graphics to visually build on familiar cases and illustrate how these same functions extend beyond the real axis. It covers several important topics that are omitted in nearly all recent texts, including techniques for analytic continuation and discussions of elliptic functions and of Wiener–Hopf methods. It also presents current advances in research, highlighting the subject’s active and fascinating frontier. The primary audience for this textbook is undergraduate students taking an introductory course on complex variables and analytic functions. It is also...

Some Topics in Complex Analysis

Autor: E. G. Phillips

Número de Páginas: 150

International Series of Monographs in Pure and Applied Mathematics, Volume 86, Some Topics in Complex Analysis deals with a variety of topics related to complex analysis. This book discusses the method of comparison, periods of an integral, generalized Joukowski transformations, and Koebe's distortion theorems. The deductions from the maximum-modulus principle, canonical products and genus of an I.F., and Weierstrass's primary factors are also reviewed. This text likewise considers Mittag-Leffler's theorem, summation of series by the calculus of residues, definition of regular functions by integrals, and Riemann zeta function. This publication is a good reference for students and specialists researching in the field of applied and pure mathematics.

NIST Handbook of Mathematical Functions Hardback and CD-ROM

Autor: Frank W. J. Olver

Número de Páginas: 968

The new standard reference on mathematical functions, replacing the classic but outdated handbook from Abramowitz and Stegun. Includes PDF version.

Algebraic Methods and Q-special Functions

Autor: Jan Felipe Van Diejen , Luc Vinet

Número de Páginas: 302

There has been revived interest in recent years in the study of special functions. Many of the latest advances in the field were inspired by the works of R. A. Askey and colleagues on basic hypergeometric series and I. G. Macdonald on orthogonal polynomials related to root systems. Significant progress was made by the use of algebraic techniques involving quantum groups, Hecke algebras, and combinatorial methods. The CRM organized a workshop for key researchers in the field to present an overview of current trends. This volume consists of the contributions to that workshop. Topics include basic hypergeometric functions, algebraic and representation-theoretic methods, combinatorics of symmetric functions, root systems, and the connections with integrable systems.