Ebook: Special Functions for Applied Scientists

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.