Note: Cover -- Half-title -- Title -- Copyright -- Dedication -- Contents -- Preface -- Preface to the first edition -- 1 Introduction and definitions -- 1.1 Introduction and Notational Conventions -- 1.2 Padé Approximants to the Exponential Function -- 1.3 Sequences and Series -- Obstacles -- 1.4 The Baker Definition, the C-Table, and Block Structure -- 1.5 Duality and Invariance -- 2 Elementary developments -- 2.1 Numerical Calculation of Padé Approximants -- 2.2 Decipherment of Singularities from Padé Approximants and Apparent Errors -- 2.3 Some Explicit Forms for Padé Denominators -- 2.4 Bigradients and Hadamard's Formula -- 3 Pade approximants and numerical methods -- 3.1 Aitken's Δ2 Method as [L/l] Padé Approximants -- 3.2 Acceleration and Over acceleration of Convergence -- 3.3 The ε-Algorithm and the η-Algorithm -- 3.4 Wynn's Identity and the ε-Algorithm -- 3.5 Common Identities and Recursion Formulas -- 3.6 Recursive Calculation of the Coefficients of Padé Approximants -- 3.7 Kronecker's Algorithm and Cordellier's Identity -- 3.8 The Q.D. Algorithm and the Root Problem -- 4 Connection with continued fractions -- 4.1 Definitions, Recursion Relations, and Computation -- 4.2 Continued Fractions Derived from Maclaurin Series -- 4.3 Various Representations of Continued Fractions -- 4.4 The Berlekamp-Massey Algorithm and an Application of It -- 4.5 Different Types of Continued Fractions -- 4.6 Examples of Continued Fractions Which Are Padé Approximants -- 4.7 Convergence of Continued Fractions -- 5 Stieltjes series and Pólya series -- 5.1 Introduction to Stieltjes Series -- 5.2 Convergence of Stieltjes Series -- 5.3 Moment Problems and Orthogonal Polynomials -- 5.4 Stieltjes Series Convergent in |z| < -- R -- 5.4.1 Hausdorff Moment Problem -- 5.4.2 Integer Moment Problem -- 5.5 Stieltjes Series with Zero Radius of Convergence. , 5.6 Hamburger Series and the Hamburger Moment Problem -- 5.7 Polya Frequency Series -- 6 Convergence theory -- 6.1 Introduction to Convergence Theory: Rows -- 6.2 de Montessus's Theorem -- 6.3 Hermite's Formula and de Montessus's Theorem -- 6.4 Uniqueness of Convergence -- 6.5 Convergence in Measure -- 6.6 Lemniscates, Capacity, and Measure -- 6.7 The Padé Conjecture -- 7 Extensions of Padé approximants -- 7.1 Multipoint Padé Approximants -- 7.2 Baker-Gammel Approximants -- 7.3 Series Analysis -- 7.4 Padé-Laurent, Padé-Fourier, and Padé-Tchebycheff Approximants -- 7.5 Laurent-Padé Approximation and Toeplitz Systems -- 7.6 Multivariable Approximants -- 8 Multiseries approximants -- 8.1 Simultaneous Padé Approximants -- 8.2 Operator Padé Approximants -- 8.3 Rectangular Matrix Padé Approximants for Minimal Partial-Realization Problems -- 8.4 Vector Padé Approximants -- 8.4.1 Functional Padé Approximants -- 8.5 Hermite-Padé Polynomials -- 8.5.1 Minimality Definitions and Uniqueness -- 8.5.2 Table Structure Results -- 8.5.3 Recursion Relations -- 8.5.4 Existence of Sequences and the Modified Minimality Definition -- 8.6 Integral and Algebraic Approximants -- 8.6.1 Monodromy Theory -- 8.6.2 Definitions and the Accuracy-through-Order Principle -- 8.6.3 Equivalence Properties -- 8.6.4 Invariance Properties -- 8.6.5 Separation Properties -- 8.6.6 Convergence Theory -- 8.6.7 Singular Index and Amplitude Computations -- 9 Connection with integral equations and quantum mechanics -- 9.1 The General Method and Finite-Rank Kernels -- 9.2 Padé Approximants and Integral Equations with Compact Kernels -- 9.3 Projection Techniques -- 9.4 Potential Scattering -- 9.5 Derivation of Padé Approximants from Variational Principles -- 9.6 An Error Bound on Padé Approximants from Variational Principles -- 9.7 Single-sign Potentials in Scattering Theory etc. , 9.8 Variational Padé Approximants -- 9.9 Singular Potentials -- 10 Connection with numerical analysis -- 10.1 Acceleration of Convergence -- 10.2 Tchebycheff's Inequalities for the Density Function -- 10.3 Collocation and the τ-method -- 10.4 Crank-Nicholson and Related Methods for the Diffusion Equation -- 10.5 Inversion of the Laplace Transform -- 10.6 Connection with Rational Approximation -- 10.6.1 The Carathéodory-Fejér Method -- 10.7 Padé Approximants for the Riccati Equation -- 11 Connection with quantum Held theory -- 11.1 Perturbed Harmonic Oscillators -- 11.1.1 The Peres Model -- 11.1.2 The Anharmonic Oscillator -- 11.2 Pion-Pion Scattering -- 11.3 Lattice-Cutoff λφ4n Euclidean Field Theory, or the Continuous-Spin Ising Model -- Appendix: A FORTRAN FUNCTION -- Bibliography -- Index.