Baker G.A., Graves-Morris P. — Pade approximants (vol. 2) :: Электронная библиотека попечительского совета мехмата МГУ

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Baker G.A., Graves-Morris P. — Pade approximants (vol. 2)

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Название: Pade approximants (vol. 2)

Авторы: Baker G.A., Graves-Morris P.

Аннотация:

The authors cover applications to statistical mechanics and critical phenomena. There are newly extended sections devoted to circuit design, matrix Pade approximation, computational methods, and integral and algebraic approximants.

Язык:

Рубрика: Математика/Анализ/Непрерывные дроби/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1981

Количество страниц: 215

Добавлена в каталог: 06.04.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID

Предметный указатель

$\eta$ -Algorithm      I: 80—84
$\varepsilon$ -Algorithm      I: 76—80 84—90
$\varepsilon$ -Algorithm for vector sequences      II: 55
$\varepsilon$ -Algorithm, generalized      II: 14—16
A-stability      II: 151
Acceleration of convergence      I: 16—17 69—90
Accuracy, numerical      I: 58—65
Accuracy-through-order      I: 1—2
Aitken’s $\Delta^2$ method      I: 69—74
Algorithms for rational interpolation      II: 6—17
Anharmonic oscillator      II: 170—171
Arzela’s theorem      I: 175
Associated continued fraction      I: 127
Asymptotic convergence      I: 200
Asymptotic expansion      I: 221
Backward recurrence method      I: 115
Baker algorithm      I: 66
Baker definition      I: 21—22 II:
Baker — Gammel approximants      II: 21—32 63
Baker — Gammel approximants, asymptotic      II: 30—31
Baker, Gammel and Wills, conjecture of      I: 286
Baker, Gammel and Wills, theorem of      I: 32—33
Beardon’s theorems      I: 237
Behte — Salpeter equation      II: 108—113
Bigradients      I: 37—42 136—137
Bigradients, polynomial      I: 39
Binomial function      I: 139 144
Biorthogonal algorithm      II: 76—79
BLOCKS      I: 24—31
C(L/M), definition of      I: 21
C-fraction      I: 127
C-table      I: 22—23 29 31
Calculation of Pade approximants, algebraic      I: 8—14 43—48 66—68
Calculation of Pade approximants, numerical      I: 61—68
Canterbury approximants      II: 41—45
Capacity      I: 274—284
Cardioid theorem      I: 151
Carleman’s criterion for Hamburger series      I: 223
Carleman’s criterion for Stieltjes series      I: 200—202
Carleman’s criterion, application of      II: 169
Cartan’s lemma      I: 274
Cauchy — Binet formula      I: 239—240
Cauchy — Jacobi problem      II: 1
Characteristic function      II: 128—132
Chebychev      see “Tchebyscheff”
Chisholm approximants      II: 41—43
Chordal metric      I: 256—263
Clenshaw — Lord algorithm      II: 58—60
Coefficient problem      I: 65 II:
Collocation      II: 138—145
Complementary error function      I: 140 144
Continued fractions      I: 103—157
Continued fractions, associated      I: 127
Continued fractions, contraction of      I: 127 147
Continued fractions, convergents of      see “Convergents”
Continued fractions, corresponding      I: 127
Continued fractions, definition      I: 104—105 107
Continued fractions, divergence condition for      I: 148
Continued fractions, elements of      I: 104 119
Continued fractions, equivalence transformation of      I: 105—107
Continued fractions, periodic      I: 117 138
Continued fractions, recurrence relations for      I: 106 109 135
Continued fractions, regular corresponding      I: 127
Continued fractions, repeating      I: 117—138
Continued fractions, summation formula      I: 115—116
Continued fractions, terminating      I: 104
Convergence in capacity      I: 274—284
Convergence in measure      I: 263—274
Convergence of row sequences      I: 238 261
Convergents      I: 104 111—112 117—119
Cordellier’s identity      I: 88
Crank — Nicholson methods      II: 145—153
Critical point methods      I: 55—57 59—61 II: 178
D(m, n), definition of      I: 162
D-log Pade approximants      I: 55 II:
Dawson integral      I: 141
Dawson integral, generalized      II: 19
de Montessus’s theorem      I: 241—252
de Montessus’s theorem, generalization of      I: 254 II:
Defect      I: 53—55 58—59
Deficiency index      I: 20
Density function, construction of      I: 180—181
Determinancy of moment problem      I: 17 179 193 197—207
Determinantal formulas for Pade approximants      I: 4—8 43—48
Determinantal identities for Stieltjes series coefficients      I: 165—166
Determinantal inequalities      I: 227 235
Determinantal inequalities for Hamburger series coefficients      I: 227 235
Determinantal inequalities for Stieltjes series coefficients      I: 208 209
Diagonal sequences      I: 236
Diffusion processes      II: 145—153
Divergence condition      I: 148 156—157
Divided differences      II: 2
Duality      I: 31 236
Equicontinuity      I: 174—176
Equicontinuous sequences      I: 210 261
Equivalence transformation      I: 105—106
Error formula for Pade approximants of Hamburger series      II: 128—129
Error formula for Pade approximants of Stieltjes      I: 185 189—193 II:
Error formula for Pade approximation      I: 6 250
Error formula for Pade approximation, from variational principles      II: 103—106
Error function      I: 141 II:
Essential singularity      I: 49
Euclidean algorithm      I: 66 134—135
Euler — Maclaurin sum formula      II: 130
Euler’s corresponding fraction      I: 137—139
Euler’s function and series      I: 17—18 200—202
Euler’s recurrence theorem      I: 106
Euler’s summation formula      I: 115—116
Existence of Pade approximants      I: 27
Exploding vacuum      II: 166—167
Exponential approximants      II: 28
Exponential function, continued fractions for      I: 139—143
Exponential function, Pade approximants for      I: 8—14
Exponential function, Saff — Varga theorems for      I: 229—233
Exponential integral, continued fraction for      I: 140 144 II:
Exponential integral, first order      I: 186
Extended Stieltjes series      see “Hamburger series”
Factorization      II: 43
Fisher approximants      II: 45
Forward recurrence method      I: 114—115
Frobenius definition      I: 20 22
Frobenius identities      I: 85 90—93
Gamma function, Binet’s formula for      I: 202—205
Gammel — Guttman — Gaunt — Joyce approximants      II: 33—36
Gammel’s counterexample      I: 285
Gaussian quadrature      II: 127—132
General C-fraction      I: 129
Gnomic theory      II: 74
Green’s functions, bound-state      II: 113
Green’s functions, bound-state, partial wave, free      II: 89
Green’s functions, bound-state, partial wave, Jost solution      II: 90
Green’s functions, bound-state, partial wave, momentum space      II: 114
Green’s functions, bound-state, partial wave, standing wave, free      II: 89
Green’s functions, bound-state, S-wave, exponential potential      II: 98
Green’s functions, bound-state, three-dimensional, free      II: 81
Green’s functions, bound-state, three-dimensional, relativistic, free      II: 108
Green’s functions, bound-state, three-dimensional, relativistic, standing wave, free      II: 109
Green’s functions, bound-state, three-dimensional, standing wave, free      II: 85
Gregory’s series      I: 78 81
Hadamard’s determinant theorem      I: 39—42 243
Hamburger functions as real J-fractions      I: 225
Hamburger moment, definition of      I: 208 222
Hamburger series, definition of      I: 208 222
Hamburger’s theorem      I: 221
Hankel determinant      I: 7 41
Hankel matrix, condition number of      I: 64—65
Hardy’s puzzle      I: 78—80
Hausdorff measure, $\alpha$ -dimensional      I: 283
Hausdorff moment problem      I: 193—196
Herglotz functions      I: 225—227

Hermite’s formula      I: 250 II:
High field expansions      II: 179
Hilbert space methods      II: 46 68—72 103—107
Homographic invariants      I: 32—33 II: 45 53
Hughes Jones approximants      II: 43—45
Hyperbolic tangent      I: 139 143—144
Hypergeometric functions      I: 141—147 197—198
Hypergeometric functions, $_1F_1(\dot)$ , Pade approximants of      I: 43 47 142—147 154 II:
Hypergeometric functions, $_2F_0(\dot)$ , Pade approximants of      I: 43 47 141—147 154—155
Hypergeometric functions, $_2F_1(\dot)$ , Pade approximants of      I: 43—47 141—147 156
Identities for neighboring approximants      I: 90—96
Inclusion regions      I: 171 206
Incomplete gamma function      I: 140 141 145 II:
Inequalities for density function      II: 132—138
Inequalities for moments      I: 186
Integer moment problem      I: 196—197
Integral equations      II: 64—79
Interlacing      I: 168—169 210 227 II:
Invariance, homographic      I: 32—33 II: 45 53
Inverse hyperbolic tangent      I: 140 144
Inverse tangent      I: 139 144 II:
J-fraction      I: 128
Jost method      II: 121—122
K-matrix      II: 85 107 110—112
Kernels, compact      II: 67—72
Kernels, compact, completely continuous      II: 67—72
Kernels, compact, finite rank      II: 65—67 72
Kronecker’s algorithm      I: 66 II:
L-stability      II: 151
Laguerre polynomials      I: 186
Laguerre’s method      II: 165
Lanczos biorthogonal method      II: 76
Lanczos T-method      II: 138—145
Laplace transform, inversion of      II: 153—155
Lattice-cutoff field theory      II: 176—178
Laurent’s theorem      I: 49
Le Roy function      II: 32
Lemniscates      I: 274—284
Lippman — Schwinger equation      II: 82—83 113
Logarithmic capacity      I: 283
Markov problem      II: 137
Matrix Pade approximants      II: 50—56
Measure, convergence in      I: 263
Meromorphic functions, convergence of sequences of      I: 259
Mittag — Leffler star      I: 50
Moment bounds      I: 186
Moment method      II: 46
Moment problems      I: 178—179
Moment, definition of Hamburger      I: 208 222
Moment, definition of Hausdorff      I: 195
Moment, definition of Stieltjes      I: 158
Moments, bounds for      I: 186 II:
Multi-index      I: 239
Multipoint Pade approximation      I: 254 II:
Multipole      I: 49
Multivariable approximants      II: 40—50
N-point Pade approximants      II: 1—31
Natural logarithm      I: 140 144
Newton interpolating polynomials      II: 2—3
Newton — Pade approximants      II: 1—31
Nuttall’s compact form      I: 16
Nuttall’s theorem      I: 269
Orthogonal polynomials      I: 82—86 209—210 255—256
Osculatory rational interpolation      II: 1—31
P-fraction      I: 130—131
Pade approximant      I: 1—288 II:
Pade denominator, definition of      I: 4
Pade equations      I: 2—3
Pade numerator, definition of      I: 6
Pade table      I: 7—8 27—31
Pade — Tchebycheff approximants      II: 56—62
Pade-Borel approximation      II: 29—30
Pade-Fourier approximants      II: 62—63
Pade-Frobenius definition      I: 20 22
Pade-Legendre series      II: 21—29
Parabola theorem for continued fractions      I: 150
Parabola theorem of Saff and Varga      I: 229—232
Paradiagonal sequences      I: 236
Peres model      II: 167
Perron’s counterexample      I: 238
Pion — Pion scattering      II: 172—176
Poles and zeros of Pade approximants      I: 31 49—59 96—102 126 166 263
Poles and zeros of Pade approximants for Hamburger functions      I: 209 227
Poles and zeros of Pade approximants for Stieltjes functions      I: 186 193
Poloids      II: 175
Polya frequency series      I: 227—235
Pommerenke’s theorem      I: 271
Potential scattering      II: 79—126
Projection techniques      II: 72—79
Prong method      II: 43—44
Prym’s function      I: 140 147
Q. D. algorithm      I: 99 125—126
Q. D. algorithm for T-fractions      II: 20
Q. D. algorithm, generalized      II: 13—14
Q. D. Table      I: 99 125—126
Quadratic approximants      II: 36—37 49
Quadrature      II: 127—132
Quantum theory, connection with      II: 79—126
Quasi-analytic functions      I: 51
Ratio method      II: 33
Rational approximation      II: 155—162
Rational interpolation      II: 1—31
Ray sequences      I: 191
Real J-fractions      I: 128 225
Real symmetric functions      I: 160—161 180
Regular C-fraction      I: 127
Reliability      I: 63 132 II:
Rhombus rule      I: 77 81
Riccati equation, Pade approximants for      II: 162—165
Riemann sphere      I: 257
Root problem      I: 96—102
Rouche’s Theorem      I: 279
Runge’s theorem      II: 158—159
S-fraction      I: 127—128 165 206
S-matrix, partial wave      II: 89 91
Saffs theorem      I: 254
Scattering theory, quantum mechanical      II: 80—92
Schwarz’s lemma      I: 189 192
Sectorial theorem of Saff and Varga      I: 229
Seidel’s theorem      I: 148—150
Sequence, acceleration of convergence of      I: 16—17 69—90
Sequence, column      I: 30
Sequence, diagonal      I: 17 32 35
Sequence, paradiagonal      I: 30
Sequence, row      I: 30
Series analysis      I: 55—57 59—61 II:
Series, acceleration of convergence of      I: 16—17 69—90
Shafer approximants      II: 36—37
Single-sign potentials      II: 106—114
Singular potentials      II: 120—126
Spherical convergence      I: 256—263
Star Identity      I: 23
Stieltjes function      I: 158 208
Stieltjes function, definition of      I: 158 221
Stieltjes function, numerical calculation of      I: 64
Stieltjes inversion formula      I: 221
Stieltjes series      I: 158 208
Stieltjes series, definition of      I: 158 221
Stieltjes series, example of      I: 161
Stieltjes series, inequalities for Pade approximants of      I: 170
Stieltjes series, S-fraction for      I: 165
Sturm sequence      I: 167
Sylvester’s theorem      I: 23 167
T-fractions      I: 138 II:
T-matrix      II: 81—82 103 172—176
T-matrix, partial wave      II: 89
T-matrix, Toeplitz matrix      I: 67
T-matrix, Totally monotone sequence      I: 195
T-matrix, Totally positive series      I: 228
T-matrix, Trudi’s theorem      I: 42

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