Baker G.A. — Essentials of Padé Approximants in Theoretical Physics :: Электронная библиотека попечительского совета мехмата МГУ

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

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Название: Essentials of Padé Approximants in Theoretical Physics

Автор: Baker G.A.

Аннотация:

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

Язык:

Рубрика: Физика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1975

Количество страниц: 306

Добавлена в каталог: 13.12.2009

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Предметный указатель

Acceleration of convergence of sequence      108—109
Arms’ and Edrei’s theorem      257—259
Baker — Gammel — Wills conjecture      188—189
Baker’s corollary      181
Beardon’s theorem      184—185
Beardon’s theorem for [L/1]      156
Bernstein’s Theorem      266
Bessel function      70—71
Bigradients      58 61
Bounds to averages      250—251
Bounds to averages on distribution functions      248—251
Capacity      192
Carleman’s inequality      227—228
Carleman’s theorem      224—225
Cartan’s lemma      174—175 194—195
Coefficients of Pade approximants, calculation of      77—79
Compact expression for Pade approximants      36
Continued fractions      42—64
Continued fractions, associated type      56
Continued fractions, convergence theorems      47—55
Continued fractions, corresponding type      56
Continued fractions, equivalence transformations      44—46
Continued fractions, equivalent type      57
Continued fractions, Euler-type      57
Continued fractions, Euler-type, even part of      45
Continued fractions, Gauss’s continued fraction      62—64
Continued fractions, general corresponding type      61
Continued fractions, Jacobi-type      56
Continued fractions, odd part of      45
Continued fractions, Pade approximants, relation to      57—58
Continued fractions, recursion formulas, fundamental      42—44
Continued fractions, Stieltjes-type      56
Continued fractions, Taylor series, relation to      55—57
Convergence in Hausdorff measure      193—205
Convergence in Hausdorff measure, convergence theorems      197—203
Convergence in Hausdorff measure, error formula      195—196
Convergence in Hausdorff measure, examples of      204—205
Convergence in the mean on Riemann sphere      203
Convergence of general sequences      166—185
Convergence of general sequences, approximant pole locations      175—180
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      134—143
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      147—149
Convergence of vertical sequences, general entire functions      155
Convergence of vertical sequences, N-point Pade approximant, extension to the      160—165
Convergence of vertical sequences, nonvertical sequences, extension to some      154—155
Convergence of vertical sequences, “smooth” entire functions      149—154
Convergence theory      119—205
Convergence uniqueness theorems      170
Convergent      43
Critical phenomena      10 12 274—279
Critical phenomena, bounds on thermodynamic limit      279
Critical phenomena, errors in, structure of      277—279
Critical phenomena, magnetic susceptibility      10 12
DeMontessus de Ballore’s theorem      134
Determinantal solutions for Pade approximants      8—9
Discovery of Pade approximants      7
Disk problem      183
Distribution of poles and zeros      186—192
Divergence series      (see Gauss’s hypergeometric function Stieltjes series
Duality theorem      112—113
Duality theorem, matrix Pade approximants      271
Duality theorem, multivariate approximants      293
electrical circuits      288—291
Epsilon algorithm      75
Error formula for Pade approximants      195—196
Error function      73
Euler’s divergent series      76 212 216—217
Existence theorem      24—25
Exponential function, Pade approximants to      11
Exponential integral      73
Gammel-Baker approximants      (see Generalized Pade approximants)
Gammel’s example      204—205
Gauss’s hypergeometric function      62—73
Gauss’s hypergeometric function, Bessel functions as confluence of      70—71
Gauss’s hypergeometric function, confluent form of      68—70
Gauss’s hypergeometric function, continued fraction for ratio of      62—64
Gauss’s hypergeometric function, divergent series      71—73
Gauss’s hypergeometric function, special cases of      64—68
Generalized Pade approximants      263—266
Geometric-mean-arithmetic-mean inequality      227—228
Hamburger problem, convergence      23
Hamburger problem, inclusion regions      246—248
Hamburger problem, scattering physics and      281—284
Hausdorff measure      (see Convergence in Hausdorff measure)
Hermite polynomials      (see Orthogonal polynomials)
Hermite’s formula      160
identities      26—41
Identities, cross ratio      28—30
Identities, expressed with Pade coefficients      34—36
Identities, five-term      33—34
Identities, three-term      30—33
Identities, triangle      31
Identities, two-term      26—28
Inequalities, fundamental      52
Infinite coefficients, series with, approximation of      267—269
Invariance properties      5 110—117
Invariance properties, argument transformations      110—111
Invariance properties, value transformations      112—113
Invariance Theorem      113
Invariance theorem, approximants, for      271—272
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      113—117
Linear fractional group, rotations of Riemann sphere equivalent to      115—117
Location of cuts      189—192
Location of cuts, numerical examples      128—131
Mandelstam representation      285—286
Matrix Pade approximants      270—273
Measure, convergence in      (see Convergence in Hausdorff measure)
Moment problem      85
Moment problem, trigonometric      93
Montessus’s theorem      134—143
Multiple angle formulas      (see Orthogonal polynomials)
Multivariate approximants      292—293
N-point Pade approximant      100—109
N-point Pade approximant, $\textit{N}$ th roots of unity, fit on      107—108
N-point Pade approximant, Cauchy — Jacobi problem      101 105—106
N-point Pade approximant, convergence of vertical sequences      160—165
N-point Pade approximant, determinantal solution      103
N-point Pade approximant, inclusion regions      244—246
N-point Pade approximant, orthogonality property of denominators      106
N-point Pade approximant, Pade problem as a special case      100—101
Noncommunative algebra, Pade approximants over      (see Matrix Pade approximants)
Normal, definition of      24
Notation for Pade approximants      7
Numerical examples      121—132
Numerical examples, asymptotic series, behavior for      127—128
Numerical examples, convergence at regular points      121—123
Numerical examples, convergence at singular points      123—126
Numerical examples, location of cuts      128—131
Orthogonal polynomials      85—89
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      85—86
Pade table      7 9—10 13—25
Pade table, block structure      19—24
Pade table, C table      13—19
Pade table, computation of      75—76
Pade table, connection between Pade tables of $\textit{f}(x)$ and $(1+\alpha x)\textit{f}(x)$       38—41
Pade table, Gragg’s example      23
Pade table, Sylvester’s determinant identity      14—16
Pade table, Taylor series      9
Pade — Borel summation procedure      287
Pade’s block theorem      20—22
Pade’s block theorem, illustration of      22
Parabola theorem      51—54
Polya freqeuency series      252—260
Polya freqeuency series, characterization of      253—255
Polya freqeuency series, convergence properties      255—260
Quadratic form, decomposition of into a sum of squares      90
Quadratic form, eigenvalue distribution for      93—99
Quadratic form, Toeplitz      93
Quantum field theory      286—287
Quotient difference algorithm      80—84
Quotient difference algorithm, rhombus rules      81
Recursion relations      74—84
Recursion relations, Baker’s algorithm      77—78
Recursion relations, coefficient problem      77—79
Recursion relations, root problem      80—84
Recursion relations, value problem      75—76
Recursion relations, Watson’s algorithm      79
Riemann sphere      113—117
Riemann sphere, chord length      117
Riemann sphere, complex plane, spherical representation of      114—115
Riemann sphere, continuity on      166
Riemann sphere, convergence in the mean on      203
Riemann sphere, convergence on      133—134
Riemann sphere, equicontinuity on      167
Riemann sphere, rotation as linear fractional transformation      115—117
Root solving, acceleration procedure for      292
Saffs theorem      163—165
Scattering physics      280—287
Scattering physics, forward scattering amplitude      281—282
Scattering physics, partial waves      282—284
Scattering physics, potential scattering      280—285
Schwarz’s lemma      173—174
Series with infinite coefficients, approximation of      267—269
Special functions, calculational procedures      292
Stieltjes expansion theorem      91
Stieltjes for N-point problem      244—246
Stieltjes integral representation      229—230
Stieltjes representation theorem      230
Stieltjes, convergence      218—233
Stieltjes, definition      209—213
Stieltjes, determinantal conditions      210—213
Stieltjes, diagonal sequences, limit of, existence      218—219
Stieltjes, dynamic dipole polarizability      291
Stieltjes, Hamburger problem      230—233 246—248
Stieltjes, inclusion region(s) for value of approximants      234—248
Stieltjes, inequalities      213—217
Stieltjes, interlacing properties      213—214
Stieltjes, Pade approximant coefficients to series coefficients relation      220
Stieltjes, scattering physics and      285
Stieltjes, series of      207—251
Stieltjes, uniqueness, conditions for      219—228
Taylor series      3—4
Taylor series, continued fractions and      55—57
Taylor series, expansion, definition of Pade approximants from      4—8
Taylor series, Pade table      9
Taylor series, values of function and      3—4
Thiele’s reciprocal difference method      105 106
Three-body problem      286
Transfinite diameter      192
Tschebycheff’s inequalities      248—251
Two-variable approximants      292—293
Uniqueness theorem      8
Unitary, of matrix Pade approximant      273
Villani’s limit theorem      268
Wallin’s corollary      154—155
Wallin’s example      204—205
Walsh’s corollary      181
Wilson’s Theorem      143—147

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