# Contents of companion volume

Pade Approximants, Part II: Extensions and Applications by George A. Baker, Jr. and Peter Graves-Morris (ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS, Volume 14)  xiii

Editor's Statement      xv

Section Editor's Foreword      xvii

Preface           xix

Chapter 1 Introduction and Definitions           1

Introduction and Notational Conventions       1

Pade Approximants to the Exponential Function       8

Sequences and Series; Obstacles    14

The Baker Definition, the C-Table, and Block Structure ... 19

Duality and Invariance            31

Chapter 2 Direct Application   43

Direct Calculation of Pade Approximants      43

Decipherment of Singularities from Pade Approximants ...   48

Apparent Errors          58

Numerical Calculation of Pade Approximants           61

Chapter 3 Pade Approximants and Numerical Methods       69

Aitken's A2 Method as [L/l] Pade Approximants       69

Acceleration and Overacceleration of Convergence 74

The e-Algorithm and the rj-Algorithm 76

Wynn's Identity and the e-Algorithm 84

Common Identities and Recursion Formulas            90

The Q.D. Algorithm and the Root Problem    96

Chapter 4 Connection with Continued Fractions       103

Definitions and the Recurrence Relation        103

Continued Fractions Derived from Maclaurin Series 108

Algebraic and Numerical Methods     113

Various Representations of Continued Fractions      117

Different Types of Continued Fractions         127

Regular Fractions for Nondegenerate Cases            127

General Fractions for Degenerate Cases      129

Viskovatov's Algorithm for the General Case            130

Continued Fractions for Special Cases         137

Examples of Continued Fractions Which are Pade Approximants   138

Convergence of Continued Fractions            147

Chapter 5 Stieltjes Series and Poly a Series 158

Introduction to Stieltjes Series            158

Convergence of Stieltjes Series         166

Moment Problems and Orthogonal Polynomials       178

Stieltjes Series Convergent in |z|</?  186

Hausdorff Moment Problem   195

Integer Moment Problem        196

Stieltjes Series with Zero Radius of Convergence    197

Hamburger Series and the Hamburger Moment Problem . . 208

Polya Frequency Series         227

Chapter 6 Convergence Theory         236

Introduction to Convergence Theory: Rows  236

de Montessus's Theorem      239

Hermite's Formula and de Montessus's Theorem    250

Uniqueness of Convergence  256

Convergence in Measure       263

Lemniscates, Capacity, and Measure           274

Appendix         289

Bibliography    294

Index for Part I and Part II      321Contents of Part II

Contents of companion volume

Pade Approximants, Part I: Basic Theory

by George A. Baker, Jr. and Peter Graves-Morris

(ENCYCLOPEDIA OF MATHEMATICS AND ITS

APPLICATIONS, Volume 13)            xi

Editor's Statement      xiii

Section Editor's Foreword      xv

Preface           xvii

Chapter 1 Extensions of Pade Approximants            1

Baker-Gammel Approximants           21

Series Analysis           32

Multivariable Approximants    40

Chapter 2 Connection with Integral Equations and Quantum

Mechanics      64

The General Method and Finite-Rank Kernels          64

Pade Approximants and Integral Equations with

Compact Kernels        67

Projection Techniques            72

Potential Scattering     79

Derivation of Pade Approximants from Variational Principles           92

An Error Bound on Pade Approximants from

Variational Principles  103

Single-Sign Potentials in Scattering Theory, etc        106

Singular Potentials      120Chapter 3 Connection with Numerical Analysis  127

Tchebycheff's Inequalities for the Density Function  132

Collocation and the T-Method            138

Crank-Nicholson and Related Methods for the

Diffusion Equation       145

Inversion of the Laplace Transform   153

Connection with Rational Approximation       155

Pade Approximants for the Riccati Equation 162

Chapter 4 Connection with Quantum Field Theory    166

Perturbed Harmonic Oscillators         166

Pion-Pion Scattering   172

Lattice-Cutoff \if>* Euclidean Field Theory, or the Continuous-Spin Ising Model      176

Bibliography    181

Index for Part I and Part II      211