Second Order Differential Equations

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.

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.

Contains problems at the end of each chapter which reinforce the material

Features solutions of the Heun equation, usually only found in more advanced monographs

Includes useful appendices on background material, including the Poincaré-Perron theory

Includes supplementary material: sn.pub/extras

Auteur
Gerhard Kristensson received his B.S. degree in mathematics and physics in 1973, and the Ph.D. degree in theoretical physics in 1979, both from the University of Göteborg, Sweden. In 1983 he was appointed Docent in theoretical physics at the University of Göteborg. During 1977-1984 he held a research position sponsored by the National Swedish Board for Technical Development (STU) and he was Lecturer at the Institute of Theoretical Physics, Göteborg from 1980-1984. In 1984-1986 he was a Visiting Scientist at the Applied Mathematical Sciences group, Ames Laboratory, Iowa State University. He held a Docent position at the Department of Electromagnetic Theory, Royal Institute of Technology, Stockholm during 1986-1989, and in 1989 he was appointed the Chair of Electromagnetic Theory at Lund Institute of Technology, Sweden. In 1992, 1997 and 2007 he was a Visiting Erskine Fellow at the Department of Mathematics, University of Canterbury, Christchurch, New Zealand. Currently, Gerhard Kristensson is a member of the Advisory Board of Inverse Problems, the Board of Editors of Wave Motion, and the Editorial and Review Board of Journal of Electromagnetic Waves and Applications and Progress in Electromagnetic Research. He is a Fellow of the Institute of Physics, and since 2006 he is the chairman of the Swedish National committee of Radio Science (SNRV) and official representative for Sweden in the International Union of Radio Science (URSI). From 1994-2005, he was the chairman of Commission B of SNRV and Official Member of URSI, Commission B for Sweden. Kristensson's major research interests are focused on wave propagation in inhomogeneous media, especially inverse scattering problems. During recent years the propagation of transient electromagnetic waves in complex media, such as dispersive anisotropic and bi-isotropic media, has been stressed. High frequency scattering methods, asymptotic expansions, optical fibers, antenna problems, and mixture formulas are also of interest, as well as radome design problems and homogenization of complex materials.

Contenu
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