Second Order Differential Equations Special Functions and Their Classification - Kristensson, Gerhard; SpringerLink (Online service) (New York, NY : Springer New York, 2010)
Second Order Differential Equations presents a classical piece of theory concerning hypergeometric special functions as solutions of second-order linear differential equations. The theory is presented in an entirely self-contained way, starting with an introduction of the solution of the second-order differential equations and then focusingon the systematic treatment and classification of these solutions. Each chapter contains a set of problems which help reinforce the theory. Some of the preliminaries are covered in appendices at the end of the book, one of which provides an introduction to Poincaré-Perron theory, and the appendix also contains a new way of analyzing the asymptomatic behavior of solutions of differential equations. This textbook is appropriate for advanced undergraduate and graduate students in Mathematics, Physics, and Engineering interested in Ordinary and Partial Differntial Equations. A solutions manual is available online.
Basic properties of the solutions -- Equations of Fuchsian type -- Equations with one to four regular singular points -- The hypergeometric differential equation -- Legendre functions and related functions -- Confluent hypergeometric functions -- Heun s differential equation.
Keywords: Mathematics; Functional equations; Functions of complex variables; Differential Equations; Functions, special; Mathematics; Ordinary Differential Equations; Special Functions; Functions of a Complex Variable; Difference and Functional Equations