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.

Kristensson, Gerhard

Adobe Acrobat 9.3.3

Springer US

United States, United States of America

New York, New York State, 2010

2010, PS, 2010

producers:
Adobe Acrobat Pro 9.3.3

Includes bibliographical references (p. 215-216) and index.

cover-large 1
front-matter 2
Second Order Differential Equations 4
Preface 8
Contents 10
fulltext 14
Chapter 1: Introduction 14
fulltext_2 16
Chapter 2: Basic properties of the solutions 16
2.1 ODE of second order 16
2.1.1 Standard forms 16
2.1.2 Classification of points 17
2.2 The Wronskian 18
2.3 Solution at a regular point 19
2.3.1 The second solution 24
2.4 Solution at a regular singular point 25
2.4.1 The indicial equation 26
2.4.2 Convergence of the solution 30
2.4.3 The second solution—exceptional case 33
2.5 Solution at a regular singular point at infinity 37
Problems 39
fulltext_3-4 42
Chapter 3: Equations of Fuchsian type 42
3.1 Regular singular point at infinity 42
3.2 Regular point at infinity 46
3.3 The displacement theorem 49
Problems 54
fulltext_4-3 56
Chapter 4: Equations with one to four regular singular points 56
4.1 ODE with one regular singular point 56
4.1.1 Regular singular point at infinity 56
4.1.2 Regular point at infinity 57
4.2 ODE with two regular singular points 58
4.2.1 One regular singular point at infinity 58
4.2.2 Regular point at infinity 59
4.2.3 Confluence 60
4.3 ODE with three regular singular points 62
4.3.1 One regular singular point at infinity 62
4.3.2 Regular point at infinity 63
4.3.2.1 Displacements 64
4.3.2.2 Substitutions 65
4.3.2.3 Connection to a regular singular point at infinity 68
4.4 ODE with four regular singular points 69
4.4.1 One regular singular point at infinity 69
4.4.2 Regular point at infinity 70
Problems 71
fulltext_5 74
Chapter 5: The hypergeometric differential equation 74
5.1 Basic properties 74
5.2 Hypergeometric series 75
5.3 Recursion and differentiation formulae 81
5.4 Kummer’s solutions to hypergeometric differential equation 85
5.5 Integral representation of F(α;β;γ;z) 90
5.6 Barnes’ integral representation 93
5.6.1 Relation between F(·; ; ·; z) and F(·; ·; ·;1 – z) 102
5.7 Quadratic transformations 104
5.8 Hypergeometric polynomials (Jacobi) 105
5.8.1 Definition of the Jacobi polynomials 106
5.8.2 Generating function 109
5.8.3 Rodrigues’ generalized function 111
5.8.4 Orthogonality 112
5.8.5 Integral representation (Schläfli) 113
5.8.6 Recursion relation 114
Problems 117
fulltext_6-7 120
Chapter 6: Legendre functions and related functions 120
6.1 Legendre functions of first and second kind 120
6.2 Integral representations 122
6.3 Associated Legendre functions 132
Problems 134
fulltext_7-8 136
Chapter 7: Confluent hypergeometric functions 136
7.1 Confluent hypergeometric functions—first kind 136
7.1.1 Bessel functions 138
7.1.2 Integral representations 139
7.1.3 Laguerre polynomials 143
7.1.4 Hermite polynomials 145
7.2 Confluent hypergeometric functions—second kind 145
7.2.1 Integral representations 146
7.2.2 Bessel functions —revisited 149
7.3 Solutions with three singular points—a summary 150
7.4 Generalized hypergeometric series 151
Problems 151
fulltext_8-6 154
Chapter 8: Heun’s differential equation 154
8.1 Basic properties 154
8.2 Power series solution 156
8.3 Polynomial solution 159
8.4 Solution in hypergeometric polynomials 160
8.4.1 Asymptotic properties of the polynomials yn(z) 161
8.4.2 Asymptotic properties of the coefficients cn 164
8.4.3 Domain of convergence 169
8.5 Confluent Heun’s equation 171
8.6 Special examples 172
8.6.1 Lamé’s differential equation 172
8.6.2 Differential equation for spheroidal functions 173
8.6.3 Mathieu’s differential equation 174
Problems 174
back-matter 176
Appendix A: The gamma function and related functions 176
A.1 The gamma function Γ(z) 176
A.2 Estimates of the gamma function 179
A.3 The Appell symbol 186
A.4 Psi (digamma) function 188
A.5 Binomial coefficient 188
A.6 The beta function B(x;y) 189
A.7 Euler–Mascheroni constant 189
Problems 190
Appendix B: Difference equations 192
B.1 Second order recursion relations 192
B.2 Poincaré–Perron theory 195
B.3 Asymptotic behavior of recursion relations 198
B.4 Estimates of some sequences and series 206
B.4.1 Riemann zeta function 207
B.4.2 The sum Σnk=1 k–1 208
B.4.3 Convergence of a sequence 210
Appendix C: Partial fractions 213
Appendix D: Circles and ellipses in the complex plane 214
D.1 Equation of the circle 214
D.1.1 Harmonic circles 215
D.2 Equation of the ellipse 216
Appendix E: Elementary and special functions 220
E.1 Hypergeometric function 2F1(α;β;γ;z) 220
E.2 Confluent functions 1F1(α;γ;z) 222
E.2.1 Error functions 223
E.3 Confluent functions 0F1(γ;z) 223
Appendix F: Notation 224
References 225
Index 227

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 focusing on the systematic treatment and classification of these solutions. -- Back Cover.
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 Poincare-Perron theory, and the appendix also contains an alternative way of analyzing the asymptomatic behavior of solutions of difference equations. -- Back Cover.
This textbook is appropriate For advanced undergraduate and graduate students in Mathematics, Physics, and Engineering interested in Ordinary and Partial Differential Equations. A solutions manual is available online at springer.com. --Back Cover.

2024-06-27