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This comprehensive review of the theory includes sample problems and a new technique for analyzing the asymptomatic behavior of differential equation solutions. It shows how to solve second-order differential equations, and how to systematically classify them.

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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.

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Preface.- Introduction.- Basic Properties of the Solutions.- Equations of the 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.- The Gamma Function and Related Functions.- Difference Equations.- Partial Fractions.- Circles and Ellipses in the Complex Plane.- Elementary and Special Functions.- Notation.- Bibliography.- Index

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From the reviews:

"Gerhard Kristensson´s book Second Order Differential Equations: Special Functions and Their Classification concerns precisely what Felix Klein called ´the central problem of the whole of modern [i.e. late nineteenth century] mathematics.´ ... The book is well-written ... . A big picture of special function relationships emerges by the end and the book has several helpful diagrams to help visualize these relationships. ... An instructor teaching from this book might do well ... ." (John D. Cook, The Mathematical Association of America, October, 2010)

"The aim of the author is to present the classification and systematics of the pertinent special functions on the basis of the solutions of the underlying differential equations. ... The book is well written and convenient to read, and its content is interesting and useful. ... it might serve for preparing seminar talks or lectures." (Wolfgang Bühring, Mathematical Reviews, Issue 2011 j)

"The aim of this textbook is to complete the existing gap in the literature related to the structure of and relationship between different special functions. ... In every chapter, many interconnections between the considered special functions are presented. Every chapter is endowed with exercises and problems." (Boris V. Loginov, Zentralblatt MATH, Vol. 1215, 2011)

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