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Baker G.A. Essentials of Padé Approximants in Theoretical Physics
Baker G.A.  Essentials of Padé Approximants in Theoretical Physics


: Essentials of Padé Approximants in Theoretical Physics

: Baker G.A.


The main purpose of this book is to present a unified account of the essential parts of our present knowledge of Pade approximants.

: en

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ed2k: ed2k stats

: 1975

: 306

: 13.12.2009

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Acceleration of convergence of sequence      108109
Arms and Edreis theorem      257259
Baker Gammel Wills conjecture      188189
Bakers corollary      181
Beardons theorem      184185
Beardons theorem for [L/1]      156
Bernsteins Theorem      266
Bessel function      7071
Bigradients      58 61
Bounds to averages      250251
Bounds to averages on distribution functions      248251
Capacity      192
Carlemans inequality      227228
Carlemans theorem      224225
Cartans lemma      174175 194195
Coefficients of Pade approximants, calculation of      7779
Compact expression for Pade approximants      36
Continued fractions      4264
Continued fractions, associated type      56
Continued fractions, convergence theorems      4755
Continued fractions, corresponding type      56
Continued fractions, equivalence transformations      4446
Continued fractions, equivalent type      57
Continued fractions, Euler-type      57
Continued fractions, Euler-type, even part of      45
Continued fractions, Gausss continued fraction      6264
Continued fractions, general corresponding type      61
Continued fractions, Jacobi-type      56
Continued fractions, odd part of      45
Continued fractions, Pade approximants, relation to      5758
Continued fractions, recursion formulas, fundamental      4244
Continued fractions, Stieltjes-type      56
Continued fractions, Taylor series, relation to      5557
Convergence in Hausdorff measure      193205
Convergence in Hausdorff measure, convergence theorems      197203
Convergence in Hausdorff measure, error formula      195196
Convergence in Hausdorff measure, examples of      204205
Convergence in the mean on Riemann sphere      203
Convergence of general sequences      166185
Convergence of general sequences, approximant pole locations      175180
Convergence of horizontal sequences      (see Convergence of vertical sequences Duality
Convergence of vertical sequences      133 165
Convergence of vertical sequences, functions with only polar singularities      134143
Convergence of vertical sequences, functions with polar singularities and smooth nonpolar singularity      143 147
Convergence of vertical sequences, functions with several smooth boundary circle singularities      147149
Convergence of vertical sequences, general entire functions      155
Convergence of vertical sequences, N-point Pade approximant, extension to the      160165
Convergence of vertical sequences, nonvertical sequences, extension to some      154155
Convergence of vertical sequences, smooth entire functions      149154
Convergence theory      119205
Convergence uniqueness theorems      170
Convergent      43
Critical phenomena      10 12 274279
Critical phenomena, bounds on thermodynamic limit      279
Critical phenomena, errors in, structure of      277279
Critical phenomena, magnetic susceptibility      10 12
DeMontessus de Ballores theorem      134
Determinantal solutions for Pade approximants      89
Discovery of Pade approximants      7
Disk problem      183
Distribution of poles and zeros      186192
Divergence series      (see Gausss hypergeometric function Stieltjes series
Duality theorem      112113
Duality theorem, matrix Pade approximants      271
Duality theorem, multivariate approximants      293
electrical circuits      288291
Epsilon algorithm      75
Error formula for Pade approximants      195196
Error function      73
Eulers divergent series      76 212 216217
Existence theorem      2425
Exponential function, Pade approximants to      11
Exponential integral      73
Gammel-Baker approximants      (see Generalized Pade approximants)
Gammels example      204205
Gausss hypergeometric function      6273
Gausss hypergeometric function, Bessel functions as confluence of      7071
Gausss hypergeometric function, confluent form of      6870
Gausss hypergeometric function, continued fraction for ratio of      6264
Gausss hypergeometric function, divergent series      7173
Gausss hypergeometric function, special cases of      6468
Generalized Pade approximants      263266
Geometric-mean-arithmetic-mean inequality      227228
Hamburger problem, convergence      23
Hamburger problem, inclusion regions      246248
Hamburger problem, scattering physics and      281284
Hausdorff measure      (see Convergence in Hausdorff measure)
Hermite polynomials      (see Orthogonal polynomials)
Hermites formula      160
identities      2641
Identities, cross ratio      2830
Identities, expressed with Pade coefficients      3436
Identities, five-term      3334
Identities, three-term      3033
Identities, triangle      31
Identities, two-term      2628
Inequalities, fundamental      52
Infinite coefficients, series with, approximation of      267269
Invariance properties      5 110117
Invariance properties, argument transformations      110111
Invariance properties, value transformations      112113
Invariance Theorem      113
Invariance theorem, approximants, for      271272
Invariance theorem, matrix Pad      6
Jost function      284
Kirchhoffs rules      289
Lagrange interpolation formula generalized      101
Lagrange Beltrami decomposition      91
Laguerre polynomials      (see Orthogonal polynomials)
Laplace transforms, inversion of      292
Legendre polynomial expansions, approximants for      291
Legendre polynomials      (see Orthogonal polynomials)
LeRoy functions      266
Linear fractional group      113117
Linear fractional group, rotations of Riemann sphere equivalent to      115117
Location of cuts      189192
Location of cuts, numerical examples      128131
Mandelstam representation      285286
Matrix Pade approximants      270273
Measure, convergence in      (see Convergence in Hausdorff measure)
Moment problem      85
Moment problem, trigonometric      93
Montessuss theorem      134143
Multiple angle formulas      (see Orthogonal polynomials)
Multivariate approximants      292293
N-point Pade approximant      100109
N-point Pade approximant, $\textit{N}$th roots of unity, fit on      107108
N-point Pade approximant, Cauchy Jacobi problem      101 105106
N-point Pade approximant, convergence of vertical sequences      160165
N-point Pade approximant, determinantal solution      103
N-point Pade approximant, inclusion regions      244246
N-point Pade approximant, orthogonality property of denominators      106
N-point Pade approximant, Pade problem as a special case      100101
Noncommunative algebra, Pade approximants over      (see Matrix Pade approximants)
Normal, definition of      24
Notation for Pade approximants      7
Numerical examples      121132
Numerical examples, asymptotic series, behavior for      127128
Numerical examples, convergence at regular points      121123
Numerical examples, convergence at singular points      123126
Numerical examples, location of cuts      128131
Orthogonal polynomials      8589
Orthogonal polynomials, extremal properties      89
Orthogonal polynomials, Hermite polynomials      88
Orthogonal polynomials, Laguerre polynomials      87
Orthogonal polynomials, Legendre polynomials      87
Orthogonal polynomials, multiple angle formulas      88
Pade denominators, orthogonality properties of      8586
Pade table      7 910 1325
Pade table, block structure      1924
Pade table, C table      1319
Pade table, computation of      7576
Pade table, connection between Pade tables of $\textit{f}(x)$ and $(1+\alpha x)\textit{f}(x)$      3841
Pade table, Graggs example      23
Pade table, Sylvesters determinant identity      1416
Pade table, Taylor series      9
Pade Borel summation procedure      287
Pades block theorem      2022
Pades block theorem, illustration of      22
Parabola theorem      5154
Polya freqeuency series      252260
Polya freqeuency series, characterization of      253255
Polya freqeuency series, convergence properties      255260
Quadratic form, decomposition of into a sum of squares      90
Quadratic form, eigenvalue distribution for      9399
Quadratic form, Toeplitz      93
Quantum field theory      286287
Quotient difference algorithm      8084
Quotient difference algorithm, rhombus rules      81
Recursion relations      7484
Recursion relations, Bakers algorithm      7778
Recursion relations, coefficient problem      7779
Recursion relations, root problem      8084
Recursion relations, value problem      7576
Recursion relations, Watsons algorithm      79
Riemann sphere      113117
Riemann sphere, chord length      117
Riemann sphere, complex plane, spherical representation of      114115
Riemann sphere, continuity on      166
Riemann sphere, convergence in the mean on      203
Riemann sphere, convergence on      133134
Riemann sphere, equicontinuity on      167
Riemann sphere, rotation as linear fractional transformation      115117
Root solving, acceleration procedure for      292
Saffs theorem      163165
Scattering physics      280287
Scattering physics, forward scattering amplitude      281282
Scattering physics, partial waves      282284
Scattering physics, potential scattering      280285
Schwarzs lemma      173174
Series with infinite coefficients, approximation of      267269
Special functions, calculational procedures      292
Stieltjes expansion theorem      91
Stieltjes for N-point problem      244246
Stieltjes integral representation      229230
Stieltjes representation theorem      230
Stieltjes, convergence      218233
Stieltjes, definition      209213
Stieltjes, determinantal conditions      210213
Stieltjes, diagonal sequences, limit of, existence      218219
Stieltjes, dynamic dipole polarizability      291
Stieltjes, Hamburger problem      230233 246248
Stieltjes, inclusion region(s) for value of approximants      234248
Stieltjes, inequalities      213217
Stieltjes, interlacing properties      213214
Stieltjes, Pade approximant coefficients to series coefficients relation      220
Stieltjes, scattering physics and      285
Stieltjes, series of      207251
Stieltjes, uniqueness, conditions for      219228
Taylor series      34
Taylor series, continued fractions and      5557
Taylor series, expansion, definition of Pade approximants from      48
Taylor series, Pade table      9
Taylor series, values of function and      34
Thieles reciprocal difference method      105 106
Three-body problem      286
Transfinite diameter      192
Tschebycheffs inequalities      248251
Two-variable approximants      292293
Uniqueness theorem      8
Unitary, of matrix Pade approximant      273
Villanis limit theorem      268
Wallins corollary      154155
Wallins example      204205
Walshs corollary      181
Wilsons Theorem      143147
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