Special Functions for Applied Scientists
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Beschreibung
Special Functions for Applied Scientists provides the required mathematical tools for researchers active in the physical sciences. The book presents a full suit of elementary functions for scholars at the PhD level and covers a wide-array of topics and begins by introducing elementary classical special functions. From there, differential equations and some applications into statistical distribution theory are examined.
The fractional calculus chapter covers fractional integrals and fractional derivatives as well as their applications to reaction-diffusion problems in physics, input-output analysis, Mittag-Leffler stochastic processes and related topics. The authors then cover q-hypergeometric functions, Ramanujan's work and Lie groups.
The latter half of this volume presents applications into stochastic processes, random variables, Mittag-Leffler processes, density estimation, order statistics, and problems in astrophysics.
Professor Dr. A.M. Mathai is Emeritus Professor of Mathematics and Statistics, McGill University, Canada.
Professor Dr. Hans J. Haubold is an astrophysicist and chief scientist at the Office of Outer Space Affairs of the United Nations.
Chapter 1 introduces elementary classical special functions. Gamma, beta, psi, zeta functions, hypergeometric functions and the associated special functions, generalizations to Meijer's G and Fox's H-functions are examined here. Discussion is confined to basic properties and selected applications. Introduction to statistical distribution theory is provided. Some recent extensions of Dirichlet integrals and Dirichlet densities are discussed. A glimpse into multivariable special functions such as Appell's functions and Lauricella functions is part of Chapter 1. Special functions as solutions of differential equations are examined. Chapter 2 is devoted to fractional calculus. Fractional integrals and fractional derivatives are discussed. Their applications to reaction-diffusion problems in physics, input-output analysis, and Mittag-Leffler stochastic processes are developed. Chapter 3 deals with q-hyper-geometric or basic hypergeometric functions. Chapter 4 covers basic hypergeometric functions and Ramanujan's work on elliptic and theta functions. Chapter 5 examines the topic of special functions and Lie groups. Chapters 6 to 9 are devoted to applications of special functions. Applications to stochastic processes, geometric infinite divisibility of random variables, Mittag-Leffler processes, alpha-Laplace processes, density estimation, order statistics and astrophysics problems, are dealt with in Chapters 6 to 9. Chapter 10 is devoted to wavelet analysis. An introduction to wavelet analysis is given. Chapter 11 deals with the Jacobians of matrix transformations. Various types of matrix transformations and the associated Jacobians are provided. Chapter 12 is devoted to the discussion of functions of matrix argument in the real case. Functions of matrix argument and the pathway models along with their applications are discussed.
Provides the required mathematical tools for researchers active in the physical sciences Presents a full suit of elementary functions for scholars at PhD level Text and exercises have been class-tested over 5 different years Includes supplementary material: sn.pub/extras
Autor*in
A.M. Mathai
Themen in »Special Functions for Applied Scientists«
Derivative Hypergeometric function Groups applications of special functions astrophysics calculus differential equation fractional calculus generalized hypergeometric functions order statistics special functions stochastic processes time series analysis
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From the reviews:
"Special functions refer to functions, of one or several real or complex variables, that play key roles in … one area of mathematics, particularly number theory, combinatorics, complex analysis, differential equations, Lie group representations, probability, statistics, and mathematical physics. … Summing Up: Recommended. Comprehensive academic, technical program, and professional collections, all levels." (D. V. Feldman, Choice, Vol. 46 (5), January, 2009)
"This book differs significant from other text books on special functions, because every chapter deals with the topic on one of the courses of the SSF. … Many examples illustrate the use of theorems and formulas. At the end of each section exercises are given. … An interesting book for all scientists interested in applications of special functions in physical sciences. It contains a speedy access to many special functions used in applications and also a lot of applications." (H.-J. Glaeske, Zentralblatt MATH, Vol. 1151, 2009)
“This book is based on selected lectures given by several speakers at the Centre for Mathematical Sciences … from 1995 to 2007. … In the present book these lectures are written up in a style for self-study by scientists, engineers and mathematicians interested in applications … . Who can profit from this book? Physicists, engineers, other scientists, applied mathematicians, economists, looking for a well-readable introduction … . this book is surely recommended to open-minded researchers in any scientific and technical area … .” (Francesco Mainardi, Journal of Approximation Theory, February, 2010)()