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Second Order Differential Equations

Special Functions and Their Classification

  • Textbook
  • © 2010

Overview

Authors:
  1. Gerhard Kristensson

    1. , Department of Electrical, Lund University, Lund, Sweden

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  • 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

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Table of contents (8 chapters)

  1. Front Matter

    Pages i-xii
    Download chapter PDF

  2. Introduction

    • Gerhard Kristensson
    Pages 1-2

  3. Basic properties of the solutions

    • Gerhard Kristensson
    Pages 3-27

  4. Equations of Fuchsian type

    • Gerhard Kristensson
    Pages 29-42

  5. Equations with one to four regular singular points

    • Gerhard Kristensson
    Pages 43-59

  6. The hypergeometric differential equation

    • Gerhard Kristensson
    Pages 61-105

  7. Legendre functions and related functions

    • Gerhard Kristensson
    Pages 107-122

  8. Confluent hypergeometric functions

    • Gerhard Kristensson
    Pages 123-139

  9. Heun’s differential equation

    • Gerhard Kristensson
    Pages 141-162

  10. Back Matter

    Pages 163-219
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Keywords

About this book

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.

Reviews

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)

Authors and Affiliations

  • , Department of Electrical, Lund University, Lund, Sweden

    Gerhard Kristensson

About the author

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.

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