Higher Special Functions : A Theory of the Central Two-Point Connection Problem Based on a Singularity Approach / Wolfgang Lay.

Format:

Book

Author/Creator:

Lay, Wolfgang, author.

Series:

Encyclopedia of mathematics and its applications

Encyclopedia of Mathematics and Its Applications Series

Language:

English

Subjects (All):

Functions, Special.

Physical Description:

1 online resource (xiv, 300 pages) : digital, PDF file(s).

Edition:

First edition.

Place of Publication:

Cambridge, England : Cambridge University Press, [2024]

Summary:

Higher special functions emerge from boundary eigenvalue problems of Fuchsian differential equations with more than three singularities. This detailed reference provides solutions for singular boundary eigenvalue problems of linear ordinary differential equations of second order, exploring previously unknown methods for finding higher special functions. Starting from the fact that it is the singularities of a differential equation that determine the local, as well as the global, behaviour of its solutions, the author develops methods that are both new and efficient and lead to functional relationships that were previously unknown. All the developments discussed are placed within their historical context, allowing the reader to trace the roots of the theory back through the work of many generations of great mathematicians. Particular attention is given to the work of George Cecil Jaffé, who laid the foundation with the calculation of the quantum mechanical energy levels of the hydrogen molecule ion

Contents:

Cover

Half-title

Series information

Title page

Imprints page

Dedication

Contents

Preface

Acknowledgements

1 Introduction

1.1 Historical Remarks

1.2 Classical Special Functions: Testing the New in a Well-Known Area

1.3 Difference Equations of Poincaré-Perron Type

1.4 Underlying Principle and Basic Method

2 Singularities in Action

2.1 The Concept of s-Ranks

2.2 Coalescence and Confluence Processes

2.3 Local Solutions

3 Fuchsian Differential Equations: The Cornerstones

3.1 Introduction

3.2 Heun Differential Equation

3.3 Classes and Sets of Heun-Type Equations

3.4 Confluent Cases of Heun's Equation

3.5 The Land Beyond Heun

4 Central Two-Point Connection Problems and Higher Special Functions

4.1 Basic Mathematical Concept: Unmasking Recessive Solutions

4.2 Methods of Solution for the Heun Class

5 Applications and Examples

5.1 Oscillations of Dislocations in Crystals: Heun

5.2 Hydrogen Molecule Ion: Single Confluent Heun

5.3 Versiera d'Agnesi: Single Confluent Heun

5.4 The Quantum Quartic Oscillator: Triconfluent Heun

5.5 The Phenomenon of Avoided Crossings: Double Confluent Heun

5.6 The Land Beyond Heun

6 Afterword

Appendix A Standard Central Two-Point Connection Problem

Appendix B Curriculum Vitae of George Cecil Jaffé

References

Index

Notes:

Title from publisher's bibliographic system (viewed on 16 May 2024).

Description based on print version record.

Includes bibliographical references and index.

ISBN:

9781009546584

1009546589

9781009128414

1009128418